# Generating function for Poincare series of dimensions of $Tor^R_i(k,k)$

Let $R=k[t_1,\ldots,t_m]$ be a polynomial ring over a field $k$ and $I=(f_1,\ldots,f_r)$ an ideal of R. The $f_i$ shall be homogeneous for the natural grading of R and of degree greater than 1. Let $S=R/I$ and consider the series

$$\sum_{i=0}^\infty \dim_k Tor^S_i(k,k) z^i = g(z)$$

In computer experiments with Macaulay2 the function $g(z)$ always came out rational in $z$.

Especially, if the polynomials $f_\nu$ are chosen "absolutely randomly" and $r \leq m$ (what counts is probably that they are forming a regular sequence in $R$) then one seems to have

$$g(z)=(1+z)^s/(1-z)^r$$

with the $r$ from above and $r+s = m$.

My questions are

1. Is it true that $g(z)$ is always rational?
2. If 1. is true, is there an algorithm to compute $g(z)$ from $R$ and the $f_i$?
3. If 1. or 2. is unknown, is it at least true, that $S$ is exactly then a polynomial ring, when $g(z)$ is a polynomial? (With $g(z)=(1+z)^n$ being the polynomial then.)

As for (3), if $g(z)$ is a polynomial then $k$ has finite projective dimension, so in the graded case $S$ would have to be a polynomial ring.