Computing the Stanley depth

TL;DR

This paper investigates bounds for the Stanley depth of certain monomial ideal quotients and confirms Stanley's Conjecture for specific intersections of irreducible monomial ideals in polynomial rings.

Contribution

It provides an upper bound for the Stanley depth of $S/(Q igcap Q')$ and proves Stanley's Conjecture for intersections of some irreducible monomial ideals.

Findings

Upper bound for Stanley depth of $S/(Q igcap Q')$

Stanley's Conjecture holds for $Q_1 igcap Q_2$ and $S/(Q_1 igcap Q_2 igcap Q_3)$ with irreducible monomial ideals

Equality of Stanley depth in certain cases

Abstract

Let $Q$ and $Q'$ be two monomial primary ideals of a polynomial algebra $S$ over a field. We give an upper bound for the Stanley depth of $S/(Q\cap Q')$ which is reached if $Q$,$Q'$ are irreducible. Also we show that Stanley's Conjecture holds for $Q_1\cap Q_2$, $S/(Q_1\cap Q_2\cap Q_3)$, $(Q_i)_i$ being some irreducible monomial ideals of $S$.