An algorithm for computing the multigraded Hilbert depth of a module

TL;DR

This paper introduces an improved, effective algorithm for computing the multigraded Hilbert depth of modules, with adaptations for Stanley depth, supported by examples that resolve open problems in the field.

Contribution

The paper develops an enhanced algorithm for multigraded Hilbert depth computation and adapts it for Stanley depth, advancing computational methods in algebra.

Findings

Successfully computes multigraded Hilbert depth for various modules.

Adapts the algorithm to compute Stanley depth in specific cases.

Resolves several open problems related to depth computations.

Abstract

A method for computing the multigraded Hilbert depth of a module was presented in [16]. In this paper we improve the method and we introduce an effective algorithm for performing the computations. In a particular case, the algorithm may also be easily adapted for computing the Stanley depth of the module. We further present interesting examples which were found with the help of an experimental implementation of the algorithm. Thus, we completely solve several open problems proposed by Herzog in [12].