# f.g. subrings of polynomial rings and their Hilbert-Poincare series

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### Dima Pasechnik

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

### pedrito...@gmail.com

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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