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
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