Bounds for the Minimum Distance Function

TL;DR

This paper investigates the asymptotic behavior of the minimum distance function for certain classes of ideals, providing bounds for its stabilization point based on algebraic properties.

Contribution

It extends the understanding of the minimum distance function's asymptotics and establishes bounds for its stabilization point for $F$-pure and square-free monomial ideals.

Findings

Bounds for the stabilization point $r_I$ are derived.

The bounds relate to the dimension and Castelnuovo--Mumford regularity.

The study applies to $F$-pure and square-free monomial ideals.

Abstract

Let $I$ be a homogeneous ideal in a polynomial ring $S$. In this paper, we extend the study of the asymptotic behavior of the minimum distance function $\delta_I$ of $I$ and give bounds for its stabilization point, $r_I$, when $I$ is an $F$-pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo--Mumford regularity of $I$.