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.

1 Answer 1


I assume you mean that I is generated by homogeneous polynomials of degree d. If so, a necessary and sufficient condition is that I be generated by a regular sequence of n polynomials of degree d. See Corollary 3.3 in Stanley (Advances in Math. 28 (1978) 57-83).

  • 5
    $\begingroup$ 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. $\endgroup$ Mar 22, 2018 at 22:14
  • $\begingroup$ I stand corrected. It's still sufficient though and probably the only answer available to date. $\endgroup$
    – pjr
    Mar 24, 2018 at 12:17

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