A New Algorithm for Discussing Gröbner Bases with Parameters

In many practical applications it is necessary to determine a Gröbner basis of an ideal of polynomials whose coefficients depend on some parameters. The main problem in this context is to obtain the distinct reduced Gröbner basis for all possible values of the parameters. Examples of this situation can be found, for instance, in constructive algebraic geometry when determining conditions for a family of curves to have singular points (see Section 11.2); in robotics, in order to determine conditions on the magnitudes of a robot configuration with certain degrees of freedom and to solve the inverse kinematics problem (see Section 11.3); in the load-flow problem for a given electrical network (Montes, 1995, 1998), (see Section 11.4); in automatic theorem proving, and so on. A direct approach to this problem can be derived applying the comprehensive Gröbner basis (CGB) algorithm of Weispfenning (1992) (see also Becker and Weispfenning, 1993). The goal of Weispfenning’s algorithm is to obtain a CGB that specializes for all possible values of the parameters. For this purpose it constructs a Gröbner system, i.e. a complete set of constructible sets over the parameters, in order to add new polynomials to the initial basis and achieve the CGB. Nevertheless, the Gröbner system derived from CGB is, in general, not simple enough for applications, as it generates more cases than necessary and leads to a complex discussion. In this paper, we present an improved algorithm, called DISPGB, that abandons the objective of obtaining a CGB and focuses its interest on the essentially different reduced Gröbner basis (Gröbner system), simplifying the general discussion. A dichotomic tree discussion is carried out using quasi-canonical representations