Monomial ideals and the failure of the Strong Lefschetz property

TL;DR

This paper establishes precise lower bounds for the Hilbert function of certain artinian monomial ideals that do not satisfy the Strong Lefschetz property, extending to other ideal classes and linking to existing classifications.

Contribution

It provides sharp lower bounds for Hilbert functions of monomial ideals failing the Strong Lefschetz property and connects these bounds to prior classification results.

Findings

Sharp lower bounds for Hilbert functions of monomial ideals

Extension of bounds to other ideal classes

Connection to classification of Lefschetz property

Abstract

We give a sharp lower bound for the Hilbert function in degree $d$ of artinian quotients $\Bbbk[x_1,\ldots,x_n]/I$ failing the Strong Lefschetz property, where $I$ is a monomial ideal generated in degree $d \geq 2$. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.