How to compute the Stanley depth of a module

TL;DR

This paper presents an algorithm to compute the Stanley depth of finitely generated multigraded modules over polynomial rings, providing new insights and solutions to open questions in the field.

Contribution

The paper introduces a novel algorithm for calculating Stanley depth and applies it to answer open questions, including examples where Stanley depth exceeds module depth.

Findings

An algorithm for computing Stanley depth of modules.

An example where Stanley depth is greater than syzygy module depth.

Reduction of the problem to lattice point existence in a polytope.

Abstract

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $\mathbb{K}[X_1, \ldots, X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $M$ has Stanley depth at least $r$ can be reduced to the question whether a certain combinatorially defined polytope $\mathscr{P}$ contains a $\mathbb{Z}^n$-lattice point.