Lech's Inequality for the Buchsbaum-Rim Multiplicity and Mixed Multiplicity

TL;DR

This paper extends an improved Lech inequality to Buchsbaum-Rim and mixed multiplicities, providing sharper bounds through reduction techniques and polynomial ring analysis, especially in low dimensions.

Contribution

It generalizes Lech's inequality for Buchsbaum-Rim and mixed multiplicities, introducing new bounds and reduction methods for the problem.

Findings

Sharper bounds for mixed multiplicity in low dimensions

Reduction to graded and polynomial ring cases

Use of complete reductions to establish inequalities

Abstract

We generalize an improved Lech bound, due to Huneke, Smirnov, and Validashti, for the Buchsbaum-Rim multiplicity and mixed multiplicity. We reduce the problem to the graded case and then to the polynomial ring case. There we use complete reductions, studied by Rees, to prove sharper bounds for the mixed multiplicity in low dimensions before proving the general case.