How to compute the multigraded Hilbert depth of a module

TL;DR

This paper introduces a new method for computing the Hilbert depth of multigraded modules by reducing the problem to finite Hilbert partitions, also enabling the computation of Stanley depth.

Contribution

It presents a novel approach to compute Hilbert and Stanley depths using Hilbert partitions, simplifying the process for multigraded modules.

Findings

Method reduces Hilbert depth computation to finite Hilbert partitions

Hilbert partitions can be used to compute Stanley depth

Provides an effective computational framework for multigraded modules

Abstract

The aim of this paper is to introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,..., X_n]$ by reducing the problem to the computation of the finite set of the new defined Hilbert partitions. Moreover, in the last section, we show that Hilbert partitions may also be used for computing the Stanley depth of the module $M$.