An efficient reduction strategy for signature-based algorithms to compute Gröbner basis

Abstract

This paper introduces a strategy for signature-based algorithms to compute Gröbner basis. In comparison with Buchberger algorithm, signature-based algorithms generate S-pairs instead of S-polynomials, and operate s -reductions instead of usual reductions which is used in Buchberger algorithm. There are two strategies for s -reductions: one is only-top reduction strategy which is the way that only leading monomials are s -reduced. The other is full reduction strategy which is the way that all monomials are s -reduced. New strategy for s -reduction of S-pairs, which we call selective-full strategy, is introduced in this paper. The idea of the strategy is that when we have computed a signature Gröbner basis, there are unnecessary elements for a minimal Gröbner basis, so candidates of elements included in a minimal Gröbner basis should be sufficiently reduced. According to the experimental result, proposed strategy is efficient for computing the reduced Gröbner basis. For computing a signature Gröbner basis, it's the most efficient or not the worst of the three strategies.