Two lower bounds for the Stanley depth of monomial ideals

Abstract

Let J?I be two monomial ideals of the polynomial ring S=K[x1,...,xn]. 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 sdepth(I/J)n-l(I/J)+1 holds. On the other hand, we show that sdepth(I/J)n-dimLI/J, where dimLI/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. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim