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

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.