A New Algorithm for Discussing Gröbner Bases with Parameters

Let F be a set of polynomials in the variables __x = x1, . . . , xnwith coefficients in R [__ a ], where R is a UFD and __ a = a1, . . . ,am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning’s comprehensive Gröbner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications.