The $j$-Multiplicity of Monomial Ideals

TL;DR

This paper characterizes the j-multiplicity of monomial ideals using normalized volumes of polytopal complexes, extending volume interpretations of multiplicities and describing epsilon-multiplicity via volume of regions.

Contribution

It provides a geometric volume-based characterization of j-multiplicity for monomial ideals, extending previous interpretations of multiplicities.

Findings

j-multiplicity equals normalized volume of a polytopal complex

Extended Teissier's volume interpretation to monomial ideals

Described epsilon-multiplicity in terms of volume of a region

Abstract

We prove a characterization of the j-multiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier's volume-theoretic interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals. We also give a description of the epsilon-multiplicity of a monomial ideal in terms of the volume of a region.