A Monomial-Oriented GVW for Computing Gr\"obner Bases

TL;DR

This paper introduces a monomial-oriented variant of the GVW algorithm for computing Gr"obner bases, focusing on labeled monomials to improve efficiency and avoid generating J-pairs.

Contribution

The paper presents the mo-GVW algorithm, a novel approach that redefines the basic elements and search strategies to enhance performance in Gr"obner basis computations.

Findings

Improved computational efficiency in Gr"obner basis calculations.

Avoidance of J-pair generation leads to faster algorithms.

Effective search and checking methods enhance practical implementation.

Abstract

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gr\"obner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The mo-GVW algorithm presents a new frame of GVW and regards {\em labeled monomials} instead of {\em labeled polynomials} as basic elements of the algorithm. Being different from the original GVW algorithm, for each labeled monomial, the mo-GVW makes efforts to find the smallest signature that can generate this monomial. The mo-GVW algorithm also avoids generating J-pairs, and uses efficient methods of searching reducers and checking criteria. Thus, the mo-GVW algorithm has a better performance during practical implementations.