An Algorithm for Computing a Minimal Comprehensive Gr\"obner\, Basis of a Parametric Polynomial System

An algorithm to generate a minimal comprehensive Gr"obner, basis of a parametric polynomial system from an arbitrary faithful comprehensive Gr"obner, system is presented. A basis of a parametric polynomial ideal is a comprehensive Gr"obner, basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gr"obner, basis of the associated specialized polynomial ideal. The key idea used in ensuring minimality is that of a polynomial being essential with respect to a comprehensive Gr"obner, basis. The essentiality check is performed by determining whether a polynomial can be covered for various specializations by other polynomials in the associated branches in a comprehensive Gr"obner, system. The algorithm has been implemented and successfully tried on many examples from the literature.