On Nov 29, Simon King wrote:
Hi Reimundo,On 2020-06-17, 'Reimundo Heluani' via sage-support <sage-support@googlegroups.com> wrote:Is there an implementation of such a thing as in the title?TL;DR: Yes. Singular does have these capabilities. I recall that these were comfortably wrapped in SageMath, but as it turns out: They aren't.
Ohh this is very cool, I have a very large ticket #29610 implementing vertex algebras, and one of the main applications of that ticket is to use Groebner bases techniques to deal with algebras of functions of arc and jet spaces. I had to explicitly deal with the super and non-super situation in completely different ways because of the lack of groebner bases for the former.
Well, in the Noetherian case this works fine. The setup I need is a non-noetherian algebra: a polynomial differential algebra, that is polynomials in x_1,...,x_n and all of their formal derivatives. So this is a polynomial algebra infinitely generated by variables x_i^{(j)} for 1 <= i <= n and 0 <= j. I need to compute Hilbert series of differential ideals, that is ideals generated by some elements of the above plus all of their derivatives. This works fine in the commutative case, since I can compute grobner bases up to arbtitrary degree and ask for the hilbert series up to that degree. But I couldn't get it to work in the super-commutative case. But now that you mention that this is actually implemented in libsingular, I'll look forward to the ticket porting this to Sage. Best, R.Note to developers: I think we should have at least two trac tickets related with Reimundo's question, and we should have these tickets opened 10 years ago already!!! 1. If A is a super-commutative algebra (which exists in Sage), then singular(A) should create a copy of A in the Singular pexpect interface, but this is broken. 2. Apparently, for ideals over A, Sage is using a generic implementation that doesn't know about Gröbner bases. It should of course use an implementation based on libsingular. Longer answer: I suppose "supercommutative" means that you have something that resembles a polynomial ring, with the difference that some of the generators anti-commute among each other, whereas some generators commute with all generators.I am facing the following design problem, and I'd love to get some pointers.Can one not find it by searching in the SageMath documentation? Anyway, I suppose it is explained in the docs how to create a SCA (SuperCommutativeAlgebra) with grading, create ideals over such algebras, and compute Gröbner bases. This is how one could search during a SageMath session: sage: search_src('supercommutative') However, the answers to that search have not really been helpful. So, perhaps the search in the documentation wasn't successful either. It points to a lot of places in "sage.categories", but not where I recall the actual implementation. Fortunately I recall that supercommutative algebra is abbreviated as sca. Hence, the following search succeeded: sage: search_def('sca', whole_word=True) It tells you that you find it (of course with documentation and examples) in sage.rings.polynomial.pluralNow the issue is to compute the graded dimension of the quotient as a formal power series.You talk about Hilbert-Poincaré series? That's already implemented in SageMath
Actually I need Hilbert-Poincaré series for my computation of modular cohomology rings of finite groups; these rings are graded-commutative (for odd prime characteristic at least) and can thus be implemented as supercommutative algebras. Here is a non-trivial example from my cohomology computations (translated to boilerplate singular): sage: names = '(x1a,x1b,x3a,x3b,y2a,y2b,y2c,y2d,y6)' sage: Igens = ['x1a*x1b', 'y2b*x1a-y2a*x1b', 'y2c*x1a-y2b*x1b-y2a*x1b', 'y2c*x1b-y2b*x1b-y2a*x1b', 'y2d*x1a-y2b*x1b+y2a*x1b', '-y2b^2+y2a*y2c-y2a*y2b', 'y2b*y2c-y2b^2-y2a*y2b', 'y2c^2+y2b^2+y2a*y2b', 'y2b^2-y2a*y2d-y2a*y2b+x1a*x3a', '-y2b*y2d-y2b^2+y2a*y2b+x1b*x3a', '-y2b*y2d-y2a*y2d+x1a*x3b', '-y2c*y2d+x1b*x3b', 'y2d*x3a-y2d^2*x1b-y2c*x3a-y2b*x3a+y2a*y2b*x1b-y2a^2*x1b', '-y2b*x3a+y2a*x3b-y2a*x3a-y2a*y2b*x1b', '-y2c*x3a+y2b*x3b-y2a^2*x1b', 'y2c*x3b+y2c*x3a-y2a*y2b*x1b-y2a^2*x1b', 'x3a*x3b-y2d*x1b*x3b+y2a*x1b*x3b-y2a*x1a*x3b+y2a*x1a*x3a'] Create the super commutative algebra: sage: R = singular.ring(3,names,'wp(1,1,3,3,2,2,2,2,6)') sage: A = singular.superCommutative(4,9) sage: A.set_ring() Create an ideal and compute its Hilbert series (note that the quotient of A by this ideal is isomorphic to the cohomology ring of the Sylow 3-subgroup of U_3(8) with coefficients in GF(3)): sage: I = singular.ideal(Igens) sage: from sage.rings.polynomial.hilbert import hilbert_poincare_series sage: HP = hilbert_poincare_series(I.twostd().lead(), (1,1,3,3,2,2,2,2,6)) Now we can read off the number of standard monomials in each (weighted) degree. Unfortunately, Singular doesn't have a function to return standard monomials in WEIGHTED degree. Hence, just as a proof of concept, I collect the standard monomials in unweighted degree up to 8, then filter and count by weighted degree. sage: SM = [] sage: for d in range(1,9): ....: SM.extend(list(I.twostd().kbase(d))) ....: sage: for d in range(1,9): ....: print(d,len([m for m in SM if m.deg()==d])) ....: 1 2 2 6 3 8 4 7 5 8 6 6 7 6 8 11 And this coincides with what the Hilbert series predicts: sage: HP (t^14 + t^13 + 3*t^12 - 3*t^10 - 7*t^9 - 3*t^8 + t^7 + 2*t^6 - 2*t^5 - 5*t^4 - 7*t^3 - 6*t^2 - 2*t - 1)/(t^3 - 1) sage: PowerSeriesRing(ZZ,'t')(HP) 1 + 2*t + 6*t^2 + 8*t^3 + 7*t^4 + 8*t^5 + 6*t^6 + 6*t^7 + 11*t^8 + 13*t^9 + 9*t^10 + 11*t^11 + 10*t^12 + 8*t^13 + 10*t^14 + 10*t^15 + 8*t^16 + 10*t^17 + 10*t^18 + 8*t^19 + O(t^20) Note that in princible Singular can compute Hilbert series, too, but it has an awkward syntax, refuses to work in non-commutative settings, and occasionally suffers from integer overflows. Best regards, Simon -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/rq0279%24ev3%241%40ciao.gmane.io.
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