An Algorithm to Check Whether a Basis of a Parametric Polynomial System is a Comprehensive Gröbner Basis and the Associated Completion Algorithm

Abstract

Given a basis of a parametric polynomial ideal, an algorithm is proposed to test whether it is a comprehensive Grobner basis or not. A basis of a parametric polynomial ideal is a comprehensive Grobner basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Grobner basis of the associated specialized polynomial ideal. In case a basis does not check to be a comprehensive Grobner basis, a completion algorithm for generating a comprehensive Grobner basis from it that is patterned after Buchberger's algorithm is proposed. Its termination is proved and its correctness is established. In contrast to other algorithms for computing a comprehensive Grobner basis which first compute a comprehensive Grobner system and then extract a comprehensive Grobner basis from it, the proposed algorithm computes a comprehensive Grobner basis directly. Further, the proposed completion algorithm always computes a minimal faithful comprehensive Grobner basis in the sense that every polynomial in the result is from the ideal as well as essential with respect to the comprehensive Grobner basis. A prototype implementation of the algorithm has been successfully tried on many examples from the literature. An interesting and somewhat surprising outcome of using the proposed algorithm is that there are example parametric ideals for which a minimal comprehensive Grobner basis computed by it is different from minimal comprehensive Grobner bases computed by other algorithms in the literature.