On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One

Isolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This article is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the eighties. This theory restricts to simple zeros, i.e., where the map has corank zero. In this article we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeros and a fast algorithm for approximating them, with quadratic convergence. In case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.