On lexicographic Groebner bases of radical ideals in dimension zero: interpolation and structure

TL;DR

This paper explores the structural properties of lexicographic Groebner bases for radical ideals of dimension zero, providing explicit interpolation formulas and a triangular decomposition algorithm.

Contribution

It introduces the first explicit interpolation formula for lexicographic Groebner bases and derives a new triangular decomposition method from these bases.

Findings

Explicit interpolation formula for lexicographic Groebner bases

Structural properties enabling easy extraction of information

A new triangular decomposition algorithm

Abstract

Due to the elimination property held by the lexicographic monomial order, the corresponding Groebner bases display strong structural properties from which meaningful informations can easily be extracted. We study these properties for radical ideals of (co)dimension zero. The proof presented relies on a combinatorial decomposition of the finite set of points whereby iterated Lagrange interpolation formulas permit to reconstruct a minimal Groebner basis. This is the first fully explicit interpolation formula for polynomials forming a lexicographic Groebner basis, from which the structure property can easily be read off. The inductive nature of the proof also yield as a byproduct a triangular decomposition algorithm from the Groebner basis.