On solving parametric polynomial systems ( extended abstract )

Many authors have contributed to the study of parametric polynomial systems , and there is a large collection of references, to name a few. Various notions have been formulated for investigating the properties of parametric polynomial systems from different aspects. Border polynomial [21, 22, 20, 1], discriminant variety [11], discriminant ideal [18], discriminant set [2] are some of those notions. For (parametric) semi-algebraic systems, methods based on cylindrical algebraic decomposotion (CAD) and its variants [5, 6, 7] are applicable. However, these methods may compute much more than what is needed for the purpose of solving. One central question in the study of parametric polynomial systems is the dependence of the solutions on the parameter values. There are different ways to express the fact that the zeros of a parametric system depends continuously on the parameters in a neighborhood of a given parameter value. The notions of a border polynomial and a discriminant variety aims at capturing the parameter values at which this dependence is not continuous. The main objective of the present work is to study the relations between the notions of a border polynomial and a discriminant variety. A second intention is to gather in a single report key results on these objects, including results previously published in [20, 1]. We stress the fact that most of our results assume that the input parametric system is triangular, since triangular decomposition methods [19, 10, 8, 16, 3] can help reducing the study of general parametric systems to the triangular case. In Section 2, we revisit the notions of a border polynomial and a discriminant variety in a unified framework. In the context of triangular parametric systems, we show that the two notions essentially coincide, see Theorem 1. In Section 3, which is dedicated to parametric algebraic systems, we compare the minimal discriminant variety of a regular chain and that of its saturated ideal. This leads us to answer the following question: among all regular chains that have the same saturated ideal as a given one, what is the best choice to make the border polynomial set minimal. Most of the results in this part were presented in [20].