On the Stanley length of monomial ideals

TL;DR

This paper investigates the Stanley length of monomial ideals, providing bounds, exact formulas in specific cases, and exploring the relationship with linear quotients.

Contribution

It offers new bounds and formulas for the Stanley length of monomial ideals, and characterizes when the Stanley length equals the number of generators.

Findings

Upper bounds for Stanley length in terms of generators

Exact formulas for two-variable or two-generator cases

Equivalence of Stanley length and number of generators for ideals with linear quotients

Abstract

Let $S=K[x_1,\ldots,x_n]$ be the ring of polynomials in $n$ variables over an arbitrary field $K$. Given a finitely generated multigraded module $M$, its Stanley length, denoted by $\operatorname{slength}(M)$, is the minimal length of a Stanley decomposition of $M$. Let $I\subset S$ be a monomial ideal, minimally generated by $m$ monomials. We give an upper bound for $\operatorname{slength}(I)$, in terms of its minimal monomial generators. Also, we give precise formulas for $\operatorname{slength}(I)$, if $n=2$ or $m=2$. Also, we show that if $I$ has linear quotients, then $\operatorname{slength}(I)=m$, and the converse holds in some special cases.