Gorenstein Hilbert Coefficients

TL;DR

This paper establishes new bounds for all coefficients of the Hilbert polynomial of graded Gorenstein algebras, relating them to shifts in their resolutions, thus extending and strengthening previous bounds for multiplicity and Cohen-Macaulay algebras.

Contribution

It provides the first comprehensive bounds for all Hilbert polynomial coefficients of Gorenstein algebras based on their minimal and maximal shifts.

Findings

Bounds are established for all Hilbert polynomial coefficients.

Bounds are analogous to multiplicity bounds and are stronger than Cohen-Macaulay bounds.

Results apply to Gorenstein algebras with quasi-pure resolutions.

Abstract

We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution of $R$ . These bounds are analogous to the bounds for the multiplicity found in \cite{S} and are stronger than the bounds for the Cohen Macaulay algebras found in \cite{HZ}.