# Gotzmann Monomials In Four Variables

## TL;DR

This paper solves the open problem of characterizing Gotzmann monomials in four variables, providing a detailed and intricate classification that extends previous knowledge limited to three variables.

## Contribution

It offers the first complete characterization of Gotzmann monomials in four-variable polynomial rings, advancing understanding of Borel-stable Gotzmann ideals.

## Key findings

- Complete characterization of Gotzmann monomials in four variables
- Extension of known results from three to four variables
- Intricate structural description of the monomials

## Abstract

It is a widely open problem to determine which monomials in the n-variable polynomial ring $K[x_1,...,x_n]$ over a field $K$ have the Gotzmann property, i.e. induce a Borel-stable Gotzmann monomial ideal. Since 2007, only the case $n \le 3$ was known. Here we solve the problem for the case $n = 4$. The solution involves a surprisingly intricate characterization.