パラメータを伴ったGröbner基底の構造的な検出について

Abstract

In this talk, we introduce a method to find a term order such that the given F, a set of polynomials with parameters is a Gröbner basis for the ideal〈F〉.This problem without parameters is called the Gröbner basis detection (GBD) and there is also its simpler problem called the structural Gröbner basis detection (SGBD). GBD can be solved by the equivalent classes of term orders computed by the affine Newton polyhedron of F, and SGBD can be reduced to the maximum matching problem of bipartite graph and linear-inequality feasibility problem. Especially for SGBD with parameters, our method divides the parameter space comprehensively, and then solves each SGBD without parameters. Moreover, we also introduce some improvements using affine Newton polyhedron and comprehensive Gröbner system over modules.