Finding the number of roots of a polynomial in a plane region using the winding number

We describe a method that computes the number of roots of a polynomial f inside a region bounded by the curve @C, with an analysis of its computational cost. It is based on the number of roots being the same as the winding number of f(@C). While the usual methods for computing the winding number involve numerical integration, in this paper we use a geometrical construction. We show its correctness without referring to global information about f (like its Lipschitz constant on @C). The analysis of its cost is based on the distance from the roots to @C, expressed using a condition number suitably defined. The method can be used in a divide-and-conquer root-finding algorithm.