Multiplier Ideals and Integral Closure of Monomial Ideals: An Analytic Approach

TL;DR

This paper presents an analytic approach to understanding monomial ideals by proving key results about their membership criteria without using resolution of singularities, employing the AM-GM inequality instead.

Contribution

It introduces a novel proof method for monomial ideal properties that avoids resolution of singularities, using elementary inequalities.

Findings

Membership criteria for monomial ideals established analytically

Avoidance of resolution of singularities in proofs

Application of AM-GM inequality as a substitute for log resolution

Abstract

Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for taking a log resolution of the monomial ideal.