Polynomial Root-Finding Algorithms and Branched Covers

Introduction. The problem of devising optimal methods for numerically approximating the roots of a polynomial has been of interest for several centuries, and is far from solved. There are numerous recent works on root-finding algorithms and their cost, for example, the work of Jenkins and Traub [JT70], Renegar [Ren87], Schönhage [Sch82], and Shub and Smale [SS85, SS86, Sma85]. This list is far from complete; the reader should refer to the aforementioned papers as well as [DH69] for more detailed background. The work in this paper is most closely related to that of Smale. Our algorithm computes an approximate factorization of a given polynomial (that is, it approximates all the roots). In constructing it, we combine global topological information about polynomials (namely, that they act as branched covers of the Riemann Sphere) with a path-lifting method for finding individual roots. Utilizing this global information enables us to use fewer operations than applying the path-lifting method to each root sequentially.