Stanley depth of monomial ideals with small number of generators

TL;DR

This paper investigates the Stanley depth of monomial ideals with few generators, proving bounds and confirming the Stanley conjecture for specific cases, especially in three variables.

Contribution

It establishes new bounds for Stanley depth related to the number of generators and confirms the Stanley conjecture for monomial ideals generated by three monomials in three variables.

Findings

equate lower bounds for epth(S/I) based on minimal generators

Confirmation of Stanley conjecture for three-generated monomial ideals

Explicit computation of epth for saturated ideals in three variables

Abstract

For a monomial ideal $I\subset S=K[x_1,...,x_n]$, we show that $\sdepth(S/I)\geq n-g(I)$, where $g(I)$ is the number of the minimal monomial generators of $I$. If $I=vI'$, where $v\in S$ is a monomial, then we see that $\sdepth(S/I)=\sdepth(S/I')$. We prove that if $I$ is a monomial ideal $I\subset S$ minimally generated by three monomials, then $I$ and $S/I$ satisfy the Stanley conjecture. Given a saturated monomial ideal $I\subset K[x_1,x_2,x_3]$ we show that $\sdepth(I)=2$. As a consequence, $\sdepth(I)\geq \sdepth(K[x_1,x_2,x_3]/I)+1$ for any monomial ideal in $I\subset K[x_1,x_2,x_3]$.