How to be sure of finding a root of a complex polynomial using Newton's method

The trouble with Newton's method for finding the roots of a complex polynomial is knowing where to start the iteration. In this paper we apply the theory of rational maps and some estimates based on distortion theorems for univalent functions to find lower bounds, depending only on the degreed, for the size of regions from which the iteration will certainly converge to a root. We can also bound the number of iterations required and we give a method that works for every polynomial and takes at most some constant timesd ²(logd)² log(d ³/) iterations to find one root to within an accuracy of .