Comprehensive Gröbner basis theory for a parametric polynomial ideal and the associated completion algorithm

Abstract

Grobner basis theory for parametric polynomial ideals is explored with the main objective of mimicking the Grobner basis theory for ideals. Given a parametric polynomial ideal, its basis is a comprehensive Grobner basis if and only if for every specialization of its parameters in a given field, the specialization of the basis is a Grobner basis of the associated specialized polynomial ideal. For various specializations of parameters, structure of specialized ideals becomes qualitatively different even though there are significant relationships as well because of finiteness properties. Key concepts foundational to Grobner basis theory are reexamined and/or further developed for the parametric case: (i) Definition of a comprehensive Grobner basis, (ii) test for a comprehensive Grobner basis, (iii) parameterized rewriting, (iv) S-polynomials among parametric polynomials, (v) completion algorithm for directly computing a comprehensive Grobner basis from a given basis of a parametric ideal. Elegant properties of Grobner bases in the classical ideal theory, such as for a fixed admissible term ordering, a unique Grobner basis can be associated with every polynomial ideal as well as that such a basis can be computed from any Grobner basis of an ideal, turn out to be a major challenge to generalize for parametric ideals; issues related to these investigations are explored. A prototype implementation of the algorithm has been successfully tried on many examples from the literature.