Signature based algorithm における𝘍₄スタイルの簡約アルゴリズムの実装について

Abstract

A signature-based algorithm (SBA) computes a Groebner basis of an ideal by attaching a module monomial called signature to an element of the ideal and by finding elements of the ideal with smaller signature first. The algorithm reduces S-polynomials by an intermediate basis as in the Buchberger algorithm. However it is difficult to directly apply a 𝘍₄-like reduction of a matrix constructed from many S-polynomials and reducers because only a restricted reduction called regular reduction is allowed. In this article we propose a vectorized polynomial reduction within the framework of the sequential SBA. We report the idea for realizing this method and we show a comparison between the new implementation and the existing SBA implementation in Risa/ Asir.