On the Stanley depth and size of monomial ideals

TL;DR

This paper introduces a recursive method to estimate the Stanley depth of monomial ideals in polynomial rings and proves a lower bound related to the size of the ideal for certain classes.

Contribution

It provides a recursive formula for lower bounds of Stanley depth and establishes a new inequality linking Stanley depth and size for specific monomial ideals.

Findings

Recursive formula for Stanley depth lower bounds

Proof of Stanley depth ≥ size for certain monomial ideals

Enhanced understanding of Stanley depth in relation to ideal size

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,...,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb{K}$. For every monomial ideal $I\subset S$, We provide a recursive formula to determine a lower bound for the Stanley depth of $S/I$. We use this formula to prove the inequality ${\rm sdepth}(S/I)\geq {\rm size}(I)$ for a particular class of monomial ideals.