An Algorithm for Computing a Minimal Comprehensive Gröbner Basis of a Parametric Polynomial System
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An algorithm to generate a minimal comprehensive Gröbner basis of a parametric polynomial system from an arbitrary faithful comprehensive Gröbner system is presented. A basis of a parametric polynomial ideal is a comprehensive Gröbner basis if and only if for every specialization of parameters in a given field, the specialization of the basis is a Gröbner 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 Gröbner 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 Gröbner system. The algorithm has been implemented and successfully tried on many examples from the literature.
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