An upper bound on stability of powers of matroidal ideals

TL;DR

This paper establishes an upper bound on the stabilization indices of associated primes and depth for powers of matroidal ideals, and provides a counterexample to a previous conjecture in the field.

Contribution

It proves that the stabilization indices are bounded by the minimum of the degree and analytic spread, and refutes a conjecture about matroidal ideal stability.

Findings

Bounded the indices of stability by min{degree, analytic spread}

Provided a counterexample to Herzog and Qureshi's conjecture

Enhanced understanding of the asymptotic behavior of matroidal ideals

Abstract

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring in $n$ variables over a field $K$ and $I$ be a matroidal ideal of degree $d$. Let $\astab(I)$ and $\dstab(I)$ be the smallest integers $l$ and $k$, for which $\Ass(I^l)$ and $\depth(R/I^k)$ stabilize, respectively. In this paper, we show that $\astab(I),\dstab(I)\leq\min\{d,\ell(I)\}$, where $\ell(I)$ is the analytic spread of $I$. Furthermore, by a counterexample we give a negative answer to the conjecture of Herzog and Qureshi \cite{HQ} about stability of matroidal ideals.