On Nov 29, Simon King wrote:
We are talking about the same notions, I have my own implementation of the super or non-super non-Noetherian case. But If I want to compute the Hilbert series, say up to order n, then I divide by the ideal generated by all n+1-derivatives. This produces a polynomial algebra that I can feed to libsingular and ask for a groebner basis. I can do this in the non-super case, but not in the super case currently. R.Hi Reimundo,On 2020-11-29, 'Reimundo Heluani' via sage-support <sage-support@googlegroups.com> wrote: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 see. But this probably means that we are talking about two different notions of supercommutative algebra. I really mean a finitely generated polynomial algebra in which some generators anti-commute among each other. For the non-Noetherian case, I am not so sure if some implementation is available. What I said about SCA in Singular *is* about Noetherian algebras.
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.I associate three things with it -- but I'm not sure if one of them helps. 1. There are differential algebras in Singular, but again they seem to be finitely generated algebras. See https://www.singular.uni-kl.de/Manual/latest/sing_2482.htm, but I haven't been able to turn this into an example using the pexpect interface. 2. There are several implementations of FreeAlgebra in Sage. One of them is based on "letterplace", which is provided by Singular. It allows you to choose non-negative integer weights and can compute Gröbner bases out to any finite degree, *BUT* it only works for weighted-homogeneous ideals (and in fact it won't even let you create an element that isn't weighted homogeneous). No idea if it is possible to formulate your problem in this very restricted setting. 3. There is a so-called InfinitePolynomialRing in Sage and it can compute so-called symmetric Gröbner bases. But probably it doesn't match your needs at all, as it is commutative, and the ideals under consideration need to be "symmetric" in the sense that your algebra has indexed series of generators x_1, x_2, x_3, ..., y_1, y_2, y_3, ..., and when you take any element of your ideal and apply any permutation of {1,2,3,...} to all indices of the generators, than you again get an element of your ideal. But I'm afraid it seems your use case isn't covered. 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/rq07f4%24o75%241%40ciao.gmane.io.
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