Monomial Gotzmann sets in a quotient by a pure power

TL;DR

This paper studies Gotzmann sets in quotients of polynomial rings by pure powers, revealing their structure and growth behavior, and connecting them to Gotzmann sets in smaller polynomial rings.

Contribution

It characterizes Gotzmann sets in quotients by pure powers, linking their structure to those in the original polynomial ring and analyzing growth patterns.

Findings

Gotzmann sets in the quotient derive from specific sets in the polynomial ring.

Partitioning monomials reveals growth conditions related to smaller polynomial rings.

Properties of minimal Hilbert function growth extend to these quotient rings.

Abstract

A homogeneous set of monomials in a quotient of the polynomial ring $S:=F[x_1, \..., x_n]$ is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient $R:=F[x_1, \..., x_n]/(x_1^a)$ arise from certain Gotzmann sets in $S$. Then we partition the monomials in a Gotzmann set in $S$ with respect to the multiplicity of $x_i$ and show that if the growth of the size of a component is larger than the size of a neighboring component, then this component is a multiple of a Gotzmann set in $F[x_1, \..., x_{i-1}, x_{i+1}, \...,x_n]$. We also adopt some properties of the minimal growth of the Hilbert function in $S$ to $R$.