Caninical 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 an ascending discriminant chain of ideals, and to the corresponding descending chain of varieties in the parameter space that characterizes the different kinds of solutions. This provides a discussion of the ideal I into canonical cases even if the tree is not completely canonical. It can also be used to obtain the a canonical comprehensive Gröbner basis. With the new algorithm, we realize in some sense the objective proposed by Weispfenning in 1992. We also prove the conjectures formulated in the previous paper.