Groebner bases for exterior algebras native to Sage

[trevorkarn]

Add a native Sage implementation to compute a Groebner basis for an ideal of a Sage ExteriorAlgebra

Depends on #32369

CC: @tscrim @simon-king-jena

Component: algebra

Keywords: groebner basis, gsoc2022, f4

Author: Trevor K. Karn, Travis Scrimshaw

Branch/Commit: 36b6b20

Reviewer: Travis Scrimshaw, Trevor K. Karn

Issue created by migration from https://trac.sagemath.org/ticket/34138

[trevorkarn]

Last 10 new commits:

b2f9f50	Fix sign error
1550f9d	Fix coboundary on basis
b3ed980	Clean up code a bit
fff9062	Add doctest
9e19f79	All tests pass
001b3c7	Add doctest
1601588	Rewrite multiplication
a34ae96	Merge branch 'u/tkarn/32369-exterior-rewrite-v2' of trac.sagemath.org:sage into exterior-main
697104c	Fix merge error
d24c5fc	Initial commit of f4

[trevorkarn]

Commit: d24c5fc

[trevorkarn]

Branch: u/tkarn/exterior-gb-34138

[trevorkarn]

Dependencies: #32369

[tscrim]

Changed commit from d24c5fc to 9accac5

[tscrim]

Changed branch from u/tkarn/exterior-gb-34138 to public/algebras/exterior_groebner-34138

[tscrim]

New commits:

e94fc2f	Merge branch 'u/tkarn/exterior-gb-34138' of https://github.com/sagemath/sagetrac-mirror into u/tkarn/exterior-gb-34138
9accac5	Another implementation of Groebner bases for the exterior algebra.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

71113fa

Fixing some little details.

[sagetrac-git]

Changed commit from 9accac5 to 71113fa

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

b0f66e3	Cythonizing the element classes.
74490ce	Adding more type declarations and renaming the Groebner file.
04dc914	Initial version of GB strategy file for exterior algebra.

[sagetrac-git]

Changed commit from 71113fa to 04dc914

[sagetrac-git]

Changed commit from 04dc914 to bf0e673

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

bf0e673

Implementing different term orders for GB.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

be68f0d

Fixing some things and tests.

[sagetrac-git]

Changed commit from bf0e673 to be68f0d

[sagetrac-git]

Changed commit from be68f0d to 43045cc

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

63787c4	Bringing more to the GB strategy classes.
43045cc	Implementing containment of ideals, mostly.

[sagetrac-git]

Changed commit from 43045cc to 6480faa

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. Last 10 new commits:

7a79608	Another implementation of Groebner bases for the exterior algebra.
c8a3766	Fixing some little details.
b62d578	Cythonizing the element classes.
d91a556	Adding more type declarations and renaming the Groebner file.
3a9d790	Initial version of GB strategy file for exterior algebra.
368491c	Implementing different term orders for GB.
f2a25e3	Fixing some things and tests.
dba5089	Bringing more to the GB strategy classes.
2362a63	Implementing containment of ideals, mostly.
6480faa	Rebase off of 9.7.beta6 and fix merge conflicts

[trevorkarn]

comment:10

All tests still pass in clifford_algebra.py and clifford_algebra_element.pyx.

[trevorkarn]

comment:11

There appears to be a bug in the following example

sage: E.<x,y> = ExteriorAlgebra(QQ)
sage: I = E.ideal([x * y - 1, x * y - x])
sage: I.groebner_basis()

where it gets stuck in an infinite while loop on line 206 of exterior_algebra_groebner.pyx. For comparison,

i2 : QQ[x,y, SkewCommutative=>true]
o2 = QQ[x, y]
o2 : PolynomialRing, 2 skew commutative variables
i3 : I = ideal( x*y - 1, x*y - x)
o3 = ideal (x*y - 1, x*y - x)
o3 : Ideal of QQ[x, y]
i4 : groebnerBasis(I)
o4 = | 1 |
                      1                1
o4 : Matrix (QQ[x, y])  <--- (QQ[x, y])

[sagetrac-git]

Changed commit from 6480faa to b3ef1f7

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e95c546	Add examples
b3ef1f7	Fix matrix indexing

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

59a30c1	Merge branch 'public/algebras/exterior_groebner-34138' of https://github.com/sagemath/sagetrac-mirror into public/algebras/exterior_groebner-34138
b6174b4	Merge branch 'u/tkarn/32369-exterior-rewrite-v2' of https://github.com/sagemath/sagetrac-mirror into public/algebras/exterior_algebra_index_set-32369
a325339	Doing some reviewer changes.
47281fa	Merge branch 'public/algebras/exterior_algebra_index_set-32369' into public/algebras/exterior_groebner-34138
692f2f5	Fixing containment and reduction of exterior algebra ideals.
6c92f18	Implementing multiplication of exterior algebra ideals.

[sagetrac-git]

Changed commit from b3ef1f7 to 6c92f18

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e7e84b7

Special casing multiplication by a term.

[sagetrac-git]

Changed commit from b227973 to e7e84b7

[tscrim]

comment:26

Here is an improvement to multiplication by a single term:

sage: %timeit a * r
2.4 µs ± 34.9 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit r * a
4.88 µs ± 76.6 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit E(2) * r
5.51 µs ± 41.2 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit r * E(2)
5.63 µs ± 63.1 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
sage: %timeit E(0) * r
1.05 µs ± 7.86 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
sage: %timeit r * E(0)
1.02 µs ± 5.09 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

versus b227973:

5.09 µs ± 64.5 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
6.43 µs ± 19.7 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
5.17 µs ± 19.9 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
8.1 µs ± 33.9 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
1.34 µs ± 6.96 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)
1.76 µs ± 13.4 ns per loop (mean ± std. dev. of 7 runs, 1,000,000 loops each)

The difference with the left versus right is because we do not choose which term to slide past the other. We can maybe address that here, along with removing the code duplication I have introduced.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

36b6b20

Doing reduction in place; being careful about duplicates.

[sagetrac-git]

Changed commit from e7e84b7 to 36b6b20

[tscrim]

comment:28

Here is an update for the GBs that does the reduction in-place, which required me to remove duplicate entries during the GB. For "small" examples, it is slower:

sage: E.<x0,x1,x2,x3,x4,x5> = ExteriorAlgebra(QQ)
sage: I = E.ideal(5*x0^2*x3+37*x1*x3*x4+32*x1*x3*x5+21*x3*x5+55*x4*x5,
....:                     39*x0*x1*x5+23*x1^2*x4+57*x1*x2*x4+56*x1*x4^2+10*x2^2+52*x3*x4*x5,
....:                     33*x0^2*x3+51*x0^2+42*x0*x3*x5+51*x1^2*x4+32*x1*x3^2+x5^3,
....:                     44*x0*x3^2+42*x1*x3+47*x1*x4^2+12*x2*x3+2*x2*x4*x5+43*x3*x4^2,
....:                     49*x0^2*x2+11*x0*x1*x2+39*x0*x3*x4+44*x0*x3*x4+54*x0*x3+45*x1^2*x4,
....:                     48*x0*x2*x3+2*x2^2*x3+59*x2^2*x5+17*x2+36*x3^3+45*x4)
sage: %time GB = I.groebner_basis()
CPU times: user 116 ms, sys: 0 ns, total: 116 ms
Wall time: 115 ms

versus before

CPU times: user 76.7 ms, sys: 299 µs, total: 77 ms
Wall time: 76.7 ms

However, I think that is just due to the extra amount of information we have to store and manipulate. I strongly believe that will get dominated (and become insignificant) once the examples get larger. The data structure for that is not very efficient and could be improved. Yet, I still haven't gotten it to be anywhere close to what M2 can do; in particular this example that took you ~7s in M2:

sage: E.<p0,p1,p2,p3,p4,p5,p6,p7> = ExteriorAlgebra(QQ)
sage: I = E.ideal(3821*p2*p3*p4*p5*p6*p7-7730*p2*p3*p4*p5*p6-164*p2*p3*p4*p5*p7-2536*p2*p3*p4*p6*p7-4321*p2*p3*p5*p6*p7-2161*p2*p4*p5*p6*p7-2188*p3*p4*p5*p6*p7-486*p2*p3*p4*p5+3491*p2*p3*p4*p6+4247*p2*p3*p5*p6+3528*p2*p4*p5*p6+2616*p3*p4*p5*p6-101*p2*p3*p4*p7+1765*p2*p3*p5*p7+258*p2*p4*p5*p7-378*p3*p4*p5*p7+1246*p2*p3*p6*p7+2320*p2*p4*p6*p7+1776*p3*p4*p6*p7+1715*p2*p5*p6*p7+728*p3*p5*p6*p7+842*p4*p5*p6*p7+69*p2*p3*p4-1660*p2*p3*p5+1863*p2*p4*p5+1520*p3*p4*p5-245*p2*p3*p6-804*p2*p4*p6-2552*p3*p4*p6-3152*p2*p5*p6+40*p3*p5*p6-1213*p4*p5*p6+270*p2*p3*p7-851*p2*p4*p7+327*p3*p4*p7-1151*p2*p5*p7+1035*p3*p5*p7-161*p4*p5*p7-230*p2*p6*p7-294*p3*p6*p7-973*p4*p6*p7-264*p5*p6*p7+874*p2*p3-2212*p2*p4+168*p3*p4+511*p2*p5-918*p3*p5-2017*p4*p5-76*p2*p6+465*p3*p6+1629*p4*p6+856*p5*p6-54*p2*p7-1355*p3*p7+227*p4*p7+77*p5*p7-220*p6*p7-696*p2+458*p3+486*p4+661*p5-650*p6+671*p7-439,
....:       -6157*p1*p3*p4*p5*p6*p7+13318*p1*p3*p4*p5*p6+5928*p1*p3*p4*p5*p7+1904*p1*p3*p4*p6*p7+2109*p1*p3*p5*p6*p7+8475*p1*p4*p5*p6*p7+2878*p3*p4*p5*p6*p7-8339*p1*p3*p4*p5-2800*p1*p3*p4*p6-9649*p1*p3*p5*p6-10964*p1*p4*p5*p6-4481*p3*p4*p5*p6+251*p1*p3*p4*p7-4245*p1*p3*p5*p7-7707*p1*p4*p5*p7-2448*p3*p4*p5*p7+1057*p1*p3*p6*p7-3605*p1*p4*p6*p7+546*p3*p4*p6*p7-3633*p1*p5*p6*p7-699*p3*p5*p6*p7-4126*p4*p5*p6*p7-730*p1*p3*p4+5519*p1*p3*p5+8168*p1*p4*p5+4366*p3*p4*p5+2847*p1*p3*p6+2058*p1*p4*p6-1416*p3*p4*p6+8004*p1*p5*p6+4740*p3*p5*p6+5361*p4*p5*p6-677*p1*p3*p7+1755*p1*p4*p7-760*p3*p4*p7+3384*p1*p5*p7+2038*p3*p5*p7+4119*p4*p5*p7+812*p1*p6*p7+11*p3*p6*p7+2022*p4*p6*p7+2642*p5*p6*p7+1276*p1*p3-1723*p1*p4+121*p3*p4-6456*p1*p5-3710*p3*p5-4525*p4*p5-2187*p1*p6-1559*p3*p6-848*p4*p6-4041*p5*p6-83*p1*p7-12*p3*p7-1180*p4*p7-2747*p5*p7-1970*p6*p7+2575*p1-161*p3+2149*p4+4294*p5+1687*p6+958*p7-1950,
....:       182*p1*p2*p4*p5*p6*p7-2824*p1*p2*p4*p5*p6-3513*p1*p2*p4*p5*p7-3386*p1*p2*p4*p6*p7-2330*p1*p2*p5*p6*p7-2838*p1*p4*p5*p6*p7+1294*p2*p4*p5*p6*p7+4764*p1*p2*p4*p5+1647*p1*p2*p4*p6+4221*p1*p2*p5*p6+814*p1*p4*p5*p6+2738*p2*p4*p5*p6+4057*p1*p2*p4*p7+2403*p1*p2*p5*p7+2552*p1*p4*p5*p7+471*p2*p4*p5*p7+448*p1*p2*p6*p7+2336*p1*p4*p6*p7+1617*p2*p4*p6*p7+2220*p1*p5*p6*p7-1543*p2*p5*p6*p7+402*p4*p5*p6*p7-5184*p1*p2*p4-3983*p1*p2*p5+44*p1*p4*p5-1327*p2*p4*p5-581*p1*p2*p6-389*p1*p4*p6-2722*p2*p4*p6+443*p1*p5*p6-2893*p2*p5*p6-154*p4*p5*p6-1277*p1*p2*p7-2018*p1*p4*p7-509*p2*p4*p7-1254*p1*p5*p7+602*p2*p5*p7-464*p4*p5*p7-647*p1*p6*p7+922*p2*p6*p7-1463*p4*p6*p7+729*p5*p6*p7+2665*p1*p2+591*p1*p4+981*p2*p4-444*p1*p5+1818*p2*p5-1985*p4*p5-1818*p1*p6+197*p2*p6+1038*p4*p6+340*p5*p6+399*p1*p7-835*p2*p7+787*p4*p7-753*p5*p7-221*p6*p7+481*p1+260*p2+1713*p4+1219*p5+794*p6+762*p7-1231,
....:       2923*p1*p2*p3*p5*p6*p7-4328*p1*p2*p3*p5*p6-3674*p1*p2*p3*p5*p7-3291*p1*p2*p3*p6*p7-4955*p1*p2*p5*p6*p7-8*p1*p3*p5*p6*p7-135*p2*p3*p5*p6*p7+7784*p1*p2*p3*p5+3471*p1*p2*p3*p6+1544*p1*p2*p5*p6+1607*p1*p3*p5*p6+1710*p2*p3*p5*p6+2434*p1*p2*p3*p7+1408*p1*p2*p5*p7-215*p1*p3*p5*p7+507*p2*p3*p5*p7+2208*p1*p2*p6*p7+1920*p1*p3*p6*p7-389*p2*p3*p6*p7+1304*p1*p5*p6*p7+2480*p2*p5*p6*p7+102*p3*p5*p6*p7-2683*p1*p2*p3-3508*p1*p2*p5-3505*p1*p3*p5-2400*p2*p3*p5-2236*p1*p2*p6-1727*p1*p3*p6-1354*p2*p3*p6-1022*p1*p5*p6-2099*p2*p5*p6-918*p3*p5*p6-495*p1*p2*p7-109*p1*p3*p7+474*p2*p3*p7+268*p1*p5*p7+1084*p2*p5*p7-190*p3*p5*p7-666*p1*p6*p7-497*p2*p6*p7+615*p3*p6*p7-912*p5*p6*p7+473*p1*p2+742*p1*p3+186*p2*p3+1021*p1*p5+2556*p2*p5+2312*p3*p5+1075*p1*p6+920*p2*p6+164*p3*p6+80*p5*p6-199*p1*p7-1270*p2*p7-1050*p3*p7-724*p5*p7+136*p6*p7+740*p1-474*p2+37*p3-1056*p5+303*p6+833*p7+736,
....:       4990*p1*p2*p3*p4*p6*p7-3067*p1*p2*p3*p4*p6-1661*p1*p2*p3*p4*p7-4064*p1*p2*p3*p6*p7-223*p1*p2*p4*p6*p7-5229*p1*p3*p4*p6*p7-4636*p2*p3*p4*p6*p7+5720*p1*p2*p3*p4+4872*p1*p2*p3*p6+1643*p1*p2*p4*p6+4536*p1*p3*p4*p6+2451*p2*p3*p4*p6+1264*p1*p2*p3*p7+70*p1*p2*p4*p7+2213*p1*p3*p4*p7+1734*p2*p3*p4*p7+1698*p1*p2*p6*p7+3799*p1*p3*p6*p7+1622*p2*p3*p6*p7+901*p1*p4*p6*p7-496*p2*p4*p6*p7+3782*p3*p4*p6*p7-5591*p1*p2*p3-1303*p1*p2*p4-6383*p1*p3*p4-2332*p2*p3*p4-3179*p1*p2*p6-6257*p1*p3*p6-3654*p2*p3*p6-1830*p1*p4*p6-1473*p2*p4*p6-3278*p3*p4*p6-1462*p1*p2*p7-1495*p1*p3*p7-468*p2*p3*p7-400*p1*p4*p7+431*p2*p4*p7-1907*p3*p4*p7-1547*p1*p6*p7-214*p2*p6*p7-1423*p3*p6*p7-1625*p4*p6*p7+5708*p1*p2+3809*p1*p3+2053*p2*p3+2824*p1*p4+1122*p2*p4+3653*p3*p4+3658*p1*p6+3001*p2*p6+3890*p3*p6+2371*p4*p6+602*p1*p7+185*p2*p7+899*p3*p7+963*p4*p7+560*p6*p7-4557*p1-3536*p2-1635*p3-2552*p4-2595*p6-207*p7+2740,
....:       -1407*p1*p2*p3*p4*p5*p7+4444*p1*p2*p3*p4*p5+2350*p1*p2*p3*p4*p7+5424*p1*p2*p3*p5*p7-2524*p1*p2*p4*p5*p7-985*p1*p3*p4*p5*p7-431*p2*p3*p4*p5*p7-2662*p1*p2*p3*p4-5342*p1*p2*p3*p5-39*p1*p2*p4*p5-2525*p1*p3*p4*p5-2650*p2*p3*p4*p5-3553*p1*p2*p3*p7-71*p1*p2*p4*p7-3268*p1*p3*p4*p7-1140*p2*p3*p4*p7-702*p1*p2*p5*p7-924*p1*p3*p5*p7-2198*p2*p3*p5*p7+4087*p1*p4*p5*p7+2709*p2*p4*p5*p7+587*p3*p4*p5*p7+968*p1*p2*p3-150*p1*p2*p4+909*p1*p3*p4+4587*p2*p3*p4+929*p1*p2*p5+1804*p1*p3*p5+2226*p2*p3*p5-916*p1*p4*p5+906*p2*p4*p5+2735*p3*p4*p5+1894*p1*p2*p7+2998*p1*p3*p7+1611*p2*p3*p7+304*p1*p4*p7-1601*p2*p4*p7+2066*p3*p4*p7-1971*p1*p5*p7-480*p2*p5*p7-500*p3*p5*p7-2617*p4*p5*p7-532*p1*p2+2016*p1*p3-2574*p2*p3+529*p1*p4-1251*p2*p4-2082*p3*p4+280*p1*p5-852*p2*p5-476*p3*p5-340*p4*p5-924*p1*p7+253*p2*p7-1090*p3*p7+170*p4*p7+1204*p5*p7-869*p1+1394*p2-264*p3+719*p4+219*p5-128*p7+506,
....:       -901*p1*p2*p3*p4*p5*p6+1805*p1*p2*p3*p4*p5-1103*p1*p2*p3*p4*p6-1746*p1*p2*p3*p5*p6-1968*p1*p2*p4*p5*p6+3957*p1*p3*p4*p5*p6+1293*p2*p3*p4*p5*p6-523*p1*p2*p3*p4-2498*p1*p2*p3*p5+693*p1*p2*p4*p5-2805*p1*p3*p4*p5-722*p2*p3*p4*p5-770*p1*p2*p3*p6+1088*p1*p2*p4*p6-232*p1*p3*p4*p6+2657*p2*p3*p4*p6+3281*p1*p2*p5*p6-1066*p1*p3*p5*p6+240*p2*p3*p5*p6-1174*p1*p4*p5*p6+1304*p2*p4*p5*p6-2070*p3*p4*p5*p6+2571*p1*p2*p3+115*p1*p2*p4+3899*p1*p3*p4-4641*p2*p3*p4-752*p1*p2*p5+1531*p1*p3*p5+1178*p2*p3*p5+11*p1*p4*p5-1144*p2*p4*p5-1701*p3*p4*p5+592*p1*p2*p6+1140*p1*p3*p6+130*p2*p3*p6+304*p1*p4*p6-2273*p2*p4*p6-1224*p3*p4*p6-2*p1*p5*p6-1090*p2*p5*p6+585*p3*p5*p6+670*p4*p5*p6-1867*p1*p2-4780*p1*p3+1079*p2*p3-2435*p1*p4+2901*p2*p4+2073*p3*p4+499*p1*p5+908*p2*p5+323*p3*p5+1631*p4*p5-966*p1*p6-315*p2*p6-481*p3*p6+759*p4*p6-595*p5*p6+3233*p1-1978*p2+729*p3-1184*p4-40*p5+446*p6+282, side='left')
sage: %time GB = I.groebner_basis(reduced=False)

[tscrim]

comment:29

My guess is M2 is doing something very special and particular with the linear algebra (and maybe some stuff in parallel). Anyways, this is as best as I can get for now.

[tscrim]

comment:30

Yay, green patchbot. :).

[tscrim]

comment:32

Thank you.

Next steps for followup tickets:

• Generalizing this implementation to work with any PBW-like basis.
• The improvements noted at the end of comment:26.
• Finding other speed improvements.
• Implement conversion to/from M2 and/or Singular (via plural) to have that as an alternative algorithm.

cc-ing Simon to let him know about this as he might be interested in it.

[simon-king-jena]

comment:33

Replying to @tscrim:

Thank you.

Next steps for followup tickets:

• Generalizing this implementation to work with any PBW-like basis.
• The improvements noted at the end of comment:26.
• Finding other speed improvements.
• Implement conversion to/from M2 and/or Singular (via plural) to have that as an alternative algorithm.

cc-ing Simon to let him know about this as he might be interested in it.

Indeed I'm interested. General question (before heaving looked at the code): What was the rationale for a "native" implementation (aka "reinventing the wheel" instead of using what (e.g.) Singular has to offer?

[tscrim]

comment:34

Replying to @simon-king-jena:

Replying to @tscrim:

cc-ing Simon to let him know about this as he might be interested in it.

Indeed I'm interested. General question (before heaving looked at the code): What was the rationale for a "native" implementation (aka "reinventing the wheel" instead of using what (e.g.) Singular has to offer?

I couldn't find a dedicated implementation of GB and the exterior algebra through some (not super extensive) Google searching. With Singular using the generic g-algebra and Plural code, I didn't think it would be particularly fast (although I have no timings to back this up) and it seems quite cumbersome to go back-and-forth with. Not everyone has Macaulay2 installed. So I figured this would be the easiest way to get this functionality and, hopefully (but unfortunately does not (yet) seem to be done), the fastest way.

[simon-king-jena]

comment:35

I may be wrong, but I thought that the main feature of "exterior algebra" is being graded-commutative. And that's provided by what Singular calls "super-commutative". And that belongs to the parts of Plural that are wrapped in Sage already.

[simon-king-jena]

comment:36

Looking at SCA (from sage.rings.polynomial.plural import SCA), it seems that ideals over super-commutative algebras are mainly dealt with generically. On the other hand, I see uses of libsingular's groeber_basis functions in the SCA-code. Strange. Anyway, it is available via libsingular, and I do use it in my cohomology code. Perhaps all what's missing is an interface, so that it can be used without directly invoking libsingular.

[tscrim]

comment:37

To build quotients using GradedCommutativeAlgebra (which uses Plural), the ideals are currently mandated to be homogeneous. Although if I pass check=False, it seems like it is doing the GB computation:

sage: E.<p0,p1,p2,p3,p4,p5,p6,p7> = GradedCommutativeAlgebra(QQ)
sage: I = E.ideal(5*x0^2*x3+37*x1*x3*x4+32*x1*x3*x5+21*x3*x5+55*x4*x5,
....:                     39*x0*x1*x5+23*x1^2*x4+57*x1*x2*x4+56*x1*x4^2+10*x2^2+52*x3*x4*x5,
....:                     33*x0^2*x3+51*x0^2+42*x0*x3*x5+51*x1^2*x4+32*x1*x3^2+x5^3,
....:                     44*x0*x3^2+42*x1*x3+47*x1*x4^2+12*x2*x3+2*x2*x4*x5+43*x3*x4^2,
....:                     49*x0^2*x2+11*x0*x1*x2+39*x0*x3*x4+44*x0*x3*x4+54*x0*x3+45*x1^2*x4,
....:                     48*x0*x2*x3+2*x2^2*x3+59*x2^2*x5+17*x2+36*x3^3+45*x4)
sage: Q = E.quotient(I, check=False)
sage: Q.gens()
(x0, x1, -45/17*x4, x3, x4, x5)

Indeed, the interface for GBs does seem to be missing from these ideals. So I cannot run timings.

The specialized multiplication for the exterior algebra is apparently faster:

sage: E.<x0,x1,x2,x3,x4,x5> = ExteriorAlgebra(QQ)
sage: r = sum(E.basis())
sage: %timeit r * r
527 µs ± 6.81 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

sage: E.<x0,x1,x2,x3,x4,x5> = GradedCommutativeAlgebra(QQ)
sage: r = sum(sum(E.basis(n)) for n in range(7))
sage: %timeit r * r
712 µs ± 5.39 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)

So it should be possible to make the version of GBs here faster. It should just require being more clever about how we do the linear algebra.

Another reason why it is good to have a Sage version is this can handle any field that Sage has, which I believe is a lot more than what Singular can do (and take advantage of the corresponding specialized linear algebra code). With some hopefully minor future tweaks, this could work over more general (skew?) commutative rings.

[tscrim]

comment:38

Also, do you know if Plural handles left/right and two-sided ideals?

[vbraun]

Changed branch from public/algebras/exterior_groebner-34138 to 36b6b20