Generic initial ideals of some monomial complete intersections in four variables

TL;DR

This paper proves that for certain monomial complete intersections in four variables over a field of characteristic zero, the generic initial ideal with respect to reverse lex order is the almost revlex ideal matching the same Hilbert function.

Contribution

It establishes that the generic initial ideal of specific monomial complete intersections in four variables is the almost revlex ideal, extending understanding of their algebraic structure.

Findings

Generic initial ideals are almost revlex ideals for the considered cases.

The result applies when at least one of the exponents is two.

The proof is specific to four-variable polynomial rings.

Abstract

Let $R = K[x_1, x_2, x_3, x_4]$ be the polynomial ring over a field of characteristic zero. For the ideal $(x_1^a, x_2^b, x_3^c, x_4^d) \subset R$, where at least one of $a$, $b$, $c$ and $d$ is equal to two, we prove that its generic initial ideal with respect to the reverse lexicographic order is the almost revlex ideal corresponding to the same Hilbert function.