The Laurent coefficients of the Hilbert series of a Gorenstein algebra

TL;DR

This paper explores the Laurent coefficients of the Hilbert series of Gorenstein algebras, reformulating Stanley's functional equation into linear constraints and deriving new relations involving Bernoulli numbers and Euler polynomials.

Contribution

It introduces a novel reformulation of the Gorenstein condition via linear constraints on Laurent coefficients and connects these to Bernoulli and Euler polynomial relations.

Findings

Reformulation of Stanley's functional equation as linear constraints.

Derivation of quadratic and cubic relations for Bernoulli numbers.

Establishment of links to Euler polynomials and symmetric number triangles.

Abstract

By a theorem of R. Stanley, a graded Cohen-Macaulay domain $A$ is Gorenstein if and only if its Hilbert series satisfies the functional equation \[ \operatorname{Hilb}_A(t^{-1})=(-1)^d t^{-a}\operatorname{Hilb}_A(t), \] where $d$ is the Krull dimension and $a$ is the a-invariant of $A$. We reformulate this functional equation in terms of an infinite system of linear constraints on the Laurent coefficients of $\operatorname{Hilb}_A(t)$ at $t=1$. The main idea consists of examining the graded algebra $\mathcal F=\bigoplus_{r\in \mathbb{Z}}\mathcal F_r$ of formal power series in the variable $x$ that fulfill the condition $\varphi(x/(x-1))=(1-x)^r\varphi(x)$. As a byproduct, we derive quadratic and cubic relations for the Bernoulli numbers. The cubic relations have a natural interpretation in terms of coefficients of the Euler polynomials. For the special case of degree $r=-(a+d)=0$, these results have been investigated previously by the authors and involved merely even Euler polynomials. A link to the work of H. W. Gould and L. Carlitz on power sums of symmetric number triangles is established.