# Which ideals have standard Hilbert series?

Let $m$ and $d$ be two positive integers.

Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and finally let $H(t)$ be the Hilbert series of $R/I$.

I say that the Hilbert series $H(t)$ is standard if there exists an integer $n$ such that $$H(t) = \frac{(1-t^d)^n}{(1-t)^m} \, .$$

My question

I am looking for a (ideally, necessary and sufficient) condition on $I$ for its Hilbert series to be standard.

An example

Consider the case $m=3$ and $d=2$. Then

• For $I=\langle x_1 x_2 \rangle$, we have $H(t) = \frac{1-t^2}{(1-t)^3}$, so $I$ satisfies the condition.
• For $I=\langle x_1 x_2 , x_1 x_3 \rangle$ we have $H(t) = \frac{t^3-2 t^2+1}{(1-t)^3}$ and this can not be put in standard form, so $I$ does not satisfy the condition.

• My theorem deals only with sufficiency, not necessity. A necessary and sufficient condition seems very unlikely in view of examples like $\langle x_1^2,x_1x_2,x_2^3\rangle$, which is not generated by a regular sequence. – Richard Stanley Mar 22 '18 at 22:14