Lcm-lattices and Stanley depth: a first computational approach

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.

arXiv commentTo appear in Experimental Math. ArXiv admin note: text overlap with arXiv:1405.3602