When can one infer degrees of generators of a ring from its hilbert series

Question

I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees $d_1,\dots,d_k$.

I want a partial converse, namely if I'm given a Hilbert series and I know the orders of the poles, when can I infer degree bounds on the generators. I'm trying the following idea:

Suppose I have a subring of $\mathbb{C}[x_1,\dots,x_k]$ such that when I mod out by all the generators, the resulting ring is $\mathbb{C}$. This means that the poles are precisely degrees of the generators and the numerator is forced to be one. However, I'm not sure if this case happens if and only if my subring is freely generated.

The subring I'm considering is generated by monomials, so it has binomial relations. I'm hoping that something like "no nilpotents" means that I can apply the above argument to mod out by the generators and get $\mathbb{C}$.

Answer 1

Your second paragraph is not accurate as stated. You need the generators to be algebraically independent. At any rate, the ring $\mathbb{C}[x^2, xy, y^2, z^4]$ has Hilbert series $1/(1-t^2)^3$, so the generator of degree four cannot be gleaned from the Hilbert series.