On the Stanley depth of powers of some classes of monomial ideals

TL;DR

This paper studies the Stanley depth of powers of sums of monomial ideals and their quotients, providing insights into their asymptotic behavior and solving specific cases like monomial complete intersections.

Contribution

It introduces formulas relating the Stanley depth of combined ideals to their components and addresses the asymptotic analysis of these depths.

Findings

Formulas for Stanley depth of powers of sums of monomial ideals.

Analysis of asymptotic behavior of Stanley depth.

Solved the case of monomial complete intersections.

Abstract

Given arbitrary monomial ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $K$, we investigate the Stanley depth of powers of the sum $I+J$, and their quotient rings, in $A\otimes_K B$ in terms of those of $I$ and $J$. Our results can be used to study the asymptotic behavior of the Stanley depth of powers of a monomial ideal. For instance, we solved the case of monomial complete intersection.