An extension of Buchberger's criteria for Groebner basis decision

TL;DR

This paper introduces an extended criterion for deciding if a polynomial basis is a Groebner basis, significantly reducing the computational effort for certain systems by improving Buchberger's criteria.

Contribution

It presents a new characterization theorem and an extended criterion that reduces the number of S-polynomials needed to verify Groebner bases in specific cases.

Findings

New criterion extends Buchberger's criteria

Reduces S-polynomial checks from m(m-1)/2 to m-1 in certain systems

Improves efficiency of Groebner basis decision process

Abstract

Two fundamental questions in the theory of Groebner bases are decision ("Is a basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is not, how do we transform it into a Groebner basis?") This paper considers the first question. It is well-known that G is a Groebner basis if and only if a certain set of polynomials (the S-polynomials) satisfy a certain property. In general there are m(m-1)/2 of these, where m is the number of polynomials in G, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Groebner bases that makes use of a new criterion that extends Buchberger's Criteria. The second is the identification of a class of polynomial systems G for which the new criterion has dramatic impact, reducing the worst-case scenario from m(m-1)/2 S-polynomials to m-1.