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

Abstract

Abstract. Isolated multiple zeroes or clusters of zeroes 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 zeroes, 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 zeroes 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.

On Location and Approximation of Clusters of Zeros of Analytic Functions

“…Besides our primary interest in the several variables case [12], we now discuss the potential applications of our methods in the field of univariate polynomial root finders. As observed in [1], [37], univariate polynomials produced by eliminating variables in multivariate polynomial systems (e.g., by means of Gröbner basis computation) are often huge and "ill-conditioned."…”

“…Yet such clusters naturally arise in many theoretical and practical situations. The results of this paper are used in [12], which deals with the location and approximation of special types of clusters of multivariate analytic maps: even when starting with polynomial maps, the algorithm of [12] needs to compute with functions that are implicitly defined, hence generally not polynomial.…”

“…Even though we deal only with one complex variable map in this paper, all the results are stated with several complex variable maps, without inducing more difficulties. These results are used in [12] for several variables. We establish a list of basic properties, such as evaluation, composition, products, differentiation, etc.…”

“…Our main motivation is the application of these methods for implicit functions computed in our extended algorithm for multivariate analytic maps [12]. There we consider maps whose Jacobian matrix has corank 1.…”

Efficient polynomial system-solving by numerical methods

“…They have been generalized to the underdetermined and overdetermined cases (i.e. f : C n → C m with m = n) [ShSm96,Ded01], analysis on manifolds [Shb93,DPM03,DeSh00], the singular case [DeSh01, GLSY05,GLSY07], diophantine aspects [Mal00, CHMP01, CMSP02, CSP03], error analysis [Mal94] and even to other operators and methods. An exhaustive bibliography would be too extensive for the narrow margins of this manuscript.…”