Jul 4, 2023, 6:40:38 AMJul 4
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
So it seems that the only missing bit is computation of a presentation of S.