Polynomial System Solving

Abstract

This chapter is mainly devoted to algorithms for solving certain special zero-dimensional polynomial systems and certain applications. In the first section, we explain a few results on Grobner bases. This enables us to decide in Section 2 whether a polynomial system is zero-dimensional. We use these results to design various algorithms for zero-dimensional systems, for instance computing the multiplication table for the quotient ring and using the multiplication table to compute information about the solutions of zero-dimensional systems. A special case is treated in details in the third section. In the fourth section, we define the univariate representations and use trace computations to express the solutions of a zero-dimensional system as rational functions of the roots of a univariate polynomial. In the fifth section, we explain how to compute the limits of bounded algebraic Puiseux series which are zeros of polynomial systems. In the sixth section, we introduce the notion of pseudocritical points and design an algorithm for finding at least one point in every semi-algebraically connected component of a bounded algebraic set, using a variant of the critical point method. We end the chapter with an algorithm computing the Euler-Poincare characteristic of an algebraic set.