A note on stability properties of powers of polymatroidal ideals

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 $\mathrm{astab}(I)=\mathrm{dstab}(I)$. Furthermore, we show that if $I$ is an almost square-free Veronese type ideal of degree $d$, then $\mathrm{astab}(I)=\mathrm{dstab}(I)=\lceil \frac{n-1}{n-d}\rceil$.

arXiv comment9 pages. to appear in BIMS. arXiv admin note: text overlap with arXiv:1803.00730