A Note on Stability Properties of Powers of Polymatroidal Ideals

Abstract

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