Two lower bounds for the Stanley depth of monomial ideals

TL;DR

This paper establishes two new lower bounds for the Stanley depth of monomial ideals using the lcm number and the order dimension of the lcm lattice, and confirms Stanley's conjecture in specific cases.

Contribution

It introduces the notions of lcm number and lcm lattice dimension to bound Stanley depth and verifies Stanley's conjecture for certain classes of ideals.

Findings

Lower bound: sdepth(I/J) ≥ n - l(I/J) + 1

Lower bound: sdepth(I/J) ≥ n - dim L_{I/J}

Stanley's conjecture holds for ideals with small lcm number or lattice dimension

Abstract

Let $J\varsubsetneq I$ be two monomial ideals of the polynomial ring $S=\mathbb{K}[x_1,\ldots,x_n]$. 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 ${\rm sdepth}(I/J)\geq n-l(I/J)+1$ 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.