Upper bounds of Hilbert coefficients and Hilbert functions

TL;DR

This paper establishes improved upper bounds for the first normalized Hilbert coefficient and Hilbert functions of m-primary ideals in Cohen-Macaulay local rings, advancing understanding of their algebraic properties.

Contribution

It introduces elementary proofs of sharper upper bounds for Hilbert coefficients and functions, extending known results for maximal ideals.

Findings

New upper bounds for the first normalized Hilbert coefficient.

Extended bounds for Hilbert functions of m-primary ideals.

Improved bounds applicable to Cohen-Macaulay local rings.

Abstract

Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that improves all known upper bounds unless for a finite number of cases. We also provide new upper bounds of the Hilbert functions of $I$ extending the known bounds for the maximal ideal.