Suppose we have a field $k$ of characteristic 0, let $I$ be an ideal of $R=k[x_1,...,x_n]$, and let $H$ be the homogenization of $I$ in $S=R[z]$. Is there any relationship between the Hilbert series of $R/I$ and the Hilbert series of $S/H$?
I know that the Hilbert series for $R/I$ can be computed via Gröbner bases, but I'm hoping to gain information about a Gröbner basis for $I$ from the Hilbert series of $R/I$. In particular, if $R/I$ has Krull dimension 0, then I would like to compute $\dim_k(R/I)$ without relying on Gröbner bases. The hope is that if the Hilbert series for $R/I$ and $S/H$ are related, then I can use some of the techniques for homogeneous ideals to work in the non-homogeneous case.