A Bivariate Preprocessing Paradigm for Buchberger-M\"oller Algorithm

TL;DR

This paper introduces a bivariate preprocessing approach that simplifies and accelerates the Buchberger-Möller algorithm by leveraging polynomial interpolation bases, especially effective over finite prime fields.

Contribution

The paper proposes a novel bivariate preprocessing paradigm that enhances the efficiency of the BM algorithm through direct computation of interpolation bases.

Findings

Significant speed-up in BM algorithm performance.

Effective over finite prime fields with many applications.

Simplifies the computation process by integrating interpolation bases.

Abstract

For the last almost three decades, since the famous Buchberger-M\"oller(BM) algorithm emerged, there has been wide interest in vanishing ideals of points and associated interpolation polynomials. Our paradigm is based on the theory of bivariate polynomial interpolation on cartesian point sets that gives us related degree reducing interpolation monomial and Newton bases directly. Since the bases are involved in the computation process as well as contained in the final output of BM algorithm, our paradigm obviously simplifies the computation and accelerates the BM process. The experiments show that the paradigm is best suited for the computation over finite prime fields that have many applications.