6 Canonical discussion of polynomial systems with parameters ∗

Given a parametric polynomial ideal I, the algorithm DISPGB, introduced by the author in 2002, builds up a binary tree describing a dichotomic discussion of the different reduced Gröbner bases depending on the values of the parameters. An improvement using a discriminant ideal to rewrite the tree was described by Manubens and the author in 2005. In this paper we describe how to iterate the use of discriminants to rebuild the tree and show that this leads to a canonical ascending discriminant chain of ideals (up to some predictable alternatives), and to the corresponding descending chain of varieties in the parameter space that characterizes the different kinds of solutions. This provides the minimal canonical discussion of the ideal I and can also be used to obtain the minimal canonical comprehensive Gröbner basis. With the new algorithm, we completely realize the objective proposed by Weispfenning in 1992.