A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

Béla Bollobás, Malte Lackmann, Dierk Schleicher

Published 2010-09-09, updated 2011-08-29Version 2

We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that intersects the basins under Newton's method of \emph{all} roots of \emph{all} (suitably normalized) complex polynomials of fixed degrees $d$, with arbitrarily high probability. This set is an efficient and universal \emph{probabilistic} set of starting points to find all roots of polynomials of degree $d$ using Newton's method; the best known \emph{deterministic} set of starting points consists of $\lceil 1.1d(\log d)^2\rceil$ points.