Uniform Lech's inequality

TL;DR

This paper proves a uniform improvement of Lech's inequality for Noetherian local rings with certain conditions, establishing a stronger bound relating multiplicity and length of ideals.

Contribution

It introduces a uniform enhancement of Lech's inequality applicable to all m-primary ideals in specific local rings, based on the ring's properties.

Findings

Established a uniform epsilon for the inequality in rings with e(\,R_{red})>1

Extended results towards bounds with fixed number of generators

Provided partial results for further improvements of Lech's inequality

Abstract

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq 2$. We prove that if $e(\widehat{R}_{red})>1$, then the classical Lech's inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\varepsilon>0$ such that $e(I)\leq d!(e(R)-\varepsilon)\ell(R/I)$ for all $\mathfrak{m}$-primary ideals $I\subseteq R$. We also obtain partial results towards improvements of Lech's inequality when we fix the number of generators of $I$.