Hilbert regularity of ZZ-graded modules over polynomial rings

TL;DR

This paper introduces the concept of Hilbert regularity for ZZ-graded modules over polynomial rings, providing an arithmetical characterization, an algorithm for computation, and connecting it to Hilbert series properties.

Contribution

It defines Hilbert regularity, links it to the positivity of a related power series, and offers an algorithm to compute it and the Hilbert depth.

Findings

Arithmetical description of Hilbert regularity.

Algorithm for computing Hilbert regularity and depth.

Connection between Hilbert regularity and positivity of a transformed Hilbert series.

Abstract

Let M be a finitely generated ZZ-graded module over the standard graded polynomial ring R=K[X_1, ..., X_n] with K a field, and let H_M(t)=Q_M(t)/(1-t)^d be the Hilbert series of M. We introduce the Hilbert regularity of M as the lowest possible value of the Castelnuovo-Mumford regularity for an R-module with Hilbert series H_M. Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of M to the smallest k such that the power series Q_M(1-t)/(1-t)^k has no negative coefficients. Finally we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an R-module.