Subalgebra membership in a polynomial algebra

[John H Palmieri]

Let R = k[x_1, x_2, ..., x_n] be a polynomial ring over a field k of characteristic p. Given elements a_1, a_2, ..., a_m and b in R, I would like to know if b is in the subalgebra generated by a_1, ..., a_m.My impression from a superficial skim of the literature (Shannon and Sweedler, https://www.sciencedirect.com/science/article/pii/S0747717188800476, for example) is that this problem has been solved using Grobner bases. Is there something already in Sage that can easily perform this test? I couldn't find any method "in_subalgebra" or "in_subring" or any obviously relevant "__contains__", or even a method for constructing a subalgebra of a polynomial ring, but maybe I'm missing something obvious.

(I don't actually care about representing b inside the subalgebra, I just want to know if it's there, in case that helps.)

--

John

[Dima Pasechnik]

On Tue, Sep 6, 2022 at 10:32 PM John H Palmieri <jhpalm...@gmail.com> wrote:
>
> Let R = k[x_1, x_2, ..., x_n] be a polynomial ring over a field k of characteristic p. Given elements a_1, a_2, ..., a_m and b in R, I would like to know if b is in the subalgebra generated by a_1, ..., a_m.My impression from a superficial skim of the literature (Shannon and Sweedler, https://www.sciencedirect.com/science/article/pii/S0747717188800476, for example) is that this problem has been solved using Grobner bases. Is there something already in Sage that can easily perform this test? I couldn't find any method "in_subalgebra" or "in_subring" or any obviously relevant "__contains__", or even a method for constructing a subalgebra of a polynomial ring, but maybe I'm missing something obvious.

in Singular it is available:
https://www.singular.uni-kl.de/Manual/4-3-1/sing_1247.htm#SEC1328
So the question is to make an interface.

>
> (I don't actually care about representing b inside the subalgebra, I just want to know if it's there, in case that helps.)
>
> --
> John
>

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[John H Palmieri]

Thanks, Dima, that's helpful. I will open a ticket; I hope this will be an easy thing for people familiar with the Singular interfaces.

--

John

[John H Palmieri]

Ticket at https://trac.sagemath.org/ticket/34502. I don't know the Singular or libsingular interface, so help is appreciated.