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Jul 4, 2023, 6:40:38 AM7/4/23

to sage-devel, sage-support, Simon King, bruno....@worc.ox.ac.uk

We're looking for the ways to deal in Sage with

finitely generated subrings S=<f_1,...,f_k> of the ring of

polynomials R[x_1,...,x_n] (R a field)

of multivariate polynomial rings and their Hilbert-Poincare series.

Once you have a presentation for S, i.e. S isomorphic to R[y_1,...,y_k]/I,

with I an ideal in appropriately graded R[y_1,...,y_k], (the latter

ring should have grading deg(y_j)=deg(f_j)) one can compute

the Hilbert series H(S,t) of S as H(S,t)=H(R[y_1,...,y_k])-H(R[y_1,...,y_k]/I),

and the terms in the RHS of the latter can be computed by Sage already.

Also, as far as I understand, Sage can compute the minimal free resolution of

the module of syzygies of S, and from the resolution the presentation can be

assembled.

So it seems that the only missing bit is computation of a presentation of S.

Any pointers?

Thanks,

Dima

finitely generated subrings S=<f_1,...,f_k> of the ring of

polynomials R[x_1,...,x_n] (R a field)

of multivariate polynomial rings and their Hilbert-Poincare series.

Once you have a presentation for S, i.e. S isomorphic to R[y_1,...,y_k]/I,

with I an ideal in appropriately graded R[y_1,...,y_k], (the latter

ring should have grading deg(y_j)=deg(f_j)) one can compute

the Hilbert series H(S,t) of S as H(S,t)=H(R[y_1,...,y_k])-H(R[y_1,...,y_k]/I),

and the terms in the RHS of the latter can be computed by Sage already.

Also, as far as I understand, Sage can compute the minimal free resolution of

the module of syzygies of S, and from the resolution the presentation can be

assembled.

So it seems that the only missing bit is computation of a presentation of S.

Any pointers?

Thanks,

Dima

Jul 6, 2023, 6:43:09 AM7/6/23

to sage-s...@googlegroups.com, Simon King, bruno....@worc.ox.ac.uk, sage-devel

Dear Dina,

In the case f_1,..,f_k is a SAGBI basis, you could probably use Lemma 6.2 in

It is not very well explained (and I just found a couple of typos; apologies), but I think that you could reproduce the idea.

Hope this helps,

Pedro

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