Gotzmann monomials in four variables

Shalom Eliahou,Vittoria Bonanzinga

doi:10.33044/revuma.v62n1a04

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.

Highlights

Let K be a field and let Rn = K[x1, . . . , xn] be the n-variable polynomial algebra over K endowed with its usual grading deg(xi) = 1 for all i
We denote by Sn ⊂ Rn the set of all monomials u = xa11 · · · xann in Rn, and by Sn,d ⊂ Sn the subset of monomials of degree deg(u) = i ai = d
Determining which homogeneous ideals are Gotzmann ideals is notoriously difficult. This will be illustrated in this paper, where our determination of all monomials u in S4 such that the ideal u is a Gotzmann ideal involves a surprisingly complicated formula

Summary

Introduction

The above result for n = 3 illustrates a general property of Gotzmann monomials, proved in [4] using Gotzmann’s persistence theorem. Our main result in this paper is the classification of all Gotzmann monomials in S4 (see Theorem 7.7); we state that a monomial u = xa1xb2xc3xt is a Gotzmann monomial in S4 if and only if t≥ Before achieving this rather intricate characterization, all the easyto-perform computer-algebraic experiments we ran in order to get a clue at it were of no help.

Background and basic notions

Predecessors and successors

The maxgen monomial of lexintervals

Findings

On the first and last variables