Symbolic-numeric methods for solving polynomial equations and applications

This tutorial gives an introductive presentation of algebraic and geometric methods for solving a polynomial system f1 = · · · = fm = 0. The first class of methods is based on the study of the quotient algebra A of the polynomial ring modulo the ideal I = (f1, . . . , fm). We show how to deduce the geometry of the solutions, from the structure of A and in particular, how solving polynomial equations reduces to eigen computations of these multiplication operators. We mention briefly two general methods for computing the normal of elements in A, used to obtain a representation of the multiplication operators. The geometric methods are based projection operations, which are closely related to the theory of resultants. We present different notions and constructions of resultants and different methods for solving systems of polynomial equations, based on these formulations. Finally, we illustrate these tools on problems coming from applications in computer vision, robotics, computational biology and signal processing.