A Method to Determine if Two Parametric Polynomial Systems Are Equal

The comprehensive Gröbner systems of parametric polynomial ideal were first introduced by Volker Weispfenning. Since then, many improvements have been made to improve these algorithms to make them useful for different applications. In contract to reduced Groebner bases, which is uniquely determined by the polynomial ideal and the term ordering, however, comprehensive Groebner systems do not have such a good property. Different algorithm may give different results even for a same parametric polynomial ideal. In order to treat this issue, we give a decision method to determine whether two comprehensive Groebner systems are equal. The polynomial ideal membership problem has been solved for the non-parametric case by the classical Groebner bases method, but there is little progress on this problem for the parametric case until now. An algorithm is given for solving this problem through computing comprehensive Groebner systems. What’s more, for two parametric polynomial ideals and a constraint over the parameters defined by a constructible set, an algorithm will be given to decide whether one ideal contains the other under the constraint.