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

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