A note on stability properties of powers of polymatroidal ideals

TL;DR

This paper investigates the stability properties of powers of matroidal ideals, establishing conditions under which the associated primes and depth stabilize, and providing explicit formulas for certain classes of ideals.

Contribution

It proves that for matroidal ideals, the stabilization of associated primes and depth coincide under specific conditions, and offers explicit formulas for almost square-free Veronese type ideals.

Findings

astab(I)=1 iff dstab(I)=1

For degree d=3, astab(I)=dstab(I)

For almost square-free Veronese ideals, astab(I)=dstab(I)=ceil((n-1)/(n-d))

Abstract

Let $I$ be a matroidal ideal of degrre $d$ of a polynomial ring $R=K[x_1,...,x_n]$, where $K$ is a field. Let astab$(I)$ and dstab$(I)$ be the smallest integer $n$ for which Ass$(I^n)$ and depth$(I^n)$ stabilize, respectively. In this paper, we show that astab$(I)=1$ if and only if dstab$(I)=1$. Moreover, we prove that if $d=3$, then ${\rm astab}(I)={\rm dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then ${\rm astab}(I)={\rm dstab}(I)=\lceil\frac{n-1}{n-d}\rceil$.