Two lower bounds for the Stanley depth of monomial ideals

Abstract

Let $JI$ be two monomial ideals of the polynomial ring $S=\mathbb{K}[x_1,\dots ,x_n]$. In this paper, we provide two lower bounds for the Stanley depth of $I/J$. On the one hand, we introduce the notion of lcm number of $I/J$, denoted by $l(I/J)$, and prove that the inequality $\mathrm{sdepth}(I/J)\ge n-l(I/J)+1$ hold. On the other hand, we show that ${\sdepth}(I/J)\geq n-\dim L_{I/J}$, where $\dim L_{I/J}$ denotes the order dimension of the lcm lattice of $I/J$. We show that $I$ and $S/I$ satisfy Stanley's conjecture, if either the lcm number of $I$ or the order dimension of the lcm lattice of $I$ is small enough. Among other results, we also prove that the Stanley--Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.

arXiv commentA preliminary version of this paper by the second author was posted on ArXiv. After some discussions with the first author, this final version emerged