Lcm-lattices and Stanley depth: a first computational approach

TL;DR

This paper uses computational experiments to verify the Stanley conjecture for monomial ideals with up to five generators, providing evidence for a longstanding conjecture in combinatorial commutative algebra.

Contribution

It demonstrates that for monomial ideals with up to five generators, the Stanley depth equals the depth, confirming the Stanley conjecture in this case and supporting Herzog's conjecture.

Findings

Stanley depth equals depth for ideals with ≤5 generators

Stanley conjecture holds in these cases

Provides computational evidence for Herzog's conjecture

Abstract

Let $\mathbb{K}$ be a field, and let $S=\mathbb{K}[X_1, ..., X_n]$ be the polynomial ring. Let $I$ be a monomial ideal of $S$ with up to 5 generators. In this paper, we present a computational experiment which allows us to prove that $\mathrm{depth}_S S/I = \mathrm{sdepth}_S S/I < \mathrm{sdepth}_S I$. This shows that the Stanley conjecture is true for $S/I$ and $I$, if $I$ can be generated by at most 5 monomials. The result also brings additional computational evidence for a conjecture made by Herzog.