Bounding the First Hilbert Coefficient

TL;DR

This paper establishes new quadratic bounds on the first Hilbert coefficient of ideals in Cohen-Macaulay local rings, comparing them with existing bounds and demonstrating cases of sharpness.

Contribution

It introduces a novel quadratic bound on the first Hilbert coefficient, expanding the understanding of its behavior in Cohen-Macaulay local rings.

Findings

New quadratic bounds on the first Hilbert coefficient.

Comparison with previous bounds showing improvements.

Examples demonstrating the sharpness of the bounds.

Abstract

This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds, and give examples to show that at least in some cases it is sharp. The techniques come largely from work of Elias, Rossi, Valla, and Vasconcelos.