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Abstract: We describe a method that computes the number of roots of a polynomial inside a region bounded by the curve , with an analysis of its computational cost. It is based on the number of roots being the same as the winding number of . 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 (like its Lipschitz constant on ). The analysis of its cost is based on the distance from the roots to , expressed using a condition number suitably defined. The method can be used in a divide-and-conquer root-finding algorithm. [Copyright &y& Elsevier]