Structure of lexicographic Groebner bases in three variables of ideals of dimension zero

TL;DR

This paper extends Lazard's 1985 structural theorem for lexicographic Groebner bases from two variables to three variables, utilizing Gianni-Kalkbrener's result to analyze radical ideals.

Contribution

It generalizes the structural understanding of Groebner bases for zero-dimensional ideals from two to three variables, building on prior foundational work.

Findings

Extended Lazard's theorem to three variables

Identified restrictions for radical ideals in three-variable case

Provided a framework for analyzing zero-dimensional ideals

Abstract

We generalize the structural theorem of Lazard in 1985, from 2 variables to 3 variables. We use the Gianni-Kalkbrener result to do this, which implies some restrictions inside which lies the case of a radical ideal.