# Cohen-Macaulay local rings with $e_1 = e + 2$

## TL;DR

This paper characterizes the Hilbert functions of Cohen-Macaulay local rings with a specific relation between their first Hilbert coefficient and multiplicity, focusing on the case where e_1 equals e plus two.

## Contribution

It provides a complete description of possible Hilbert functions for Cohen-Macaulay local rings with e_1 = e + 2, a case not fully understood before.

## Key findings

- Classifies Hilbert functions for the case e_1 = e + 2
- Identifies constraints on the structure of such rings
- Advances understanding of Cohen-Macaulay ring invariants

## Abstract

In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.