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.

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    $\begingroup$ Sorry for asking a stupid question, but if $I$ is not homogeneous then what is the Hilbert series of $R/I$? $\endgroup$ Jan 18 '20 at 1:34
  • $\begingroup$ Good question! The first answer is that I had just been using Macaulay2 to compute this, so I didn’t know off the top of my head. The second answer is that $R/I$ is filtered by degree, and the Hilbert series is the Hilbert series of the associated graded $\mathrm{gr}(R/I)$. $\endgroup$ Jan 18 '20 at 5:11
  • $\begingroup$ When I was double checking this definition, I saw a post that mentioned that the Hilbert series of $R/I$ is the same as the Hilbert series of $R/\mathrm{in}(I)$, but it feels like I would again need a Gröbner basis to exploit this. $\endgroup$ Jan 18 '20 at 5:14

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