Polynomial root-finding algorithms and branched covers

Abstract

We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that computes an $ε$-factorization of the polynomial which has an arithmetic complexity of $\Order{d^2(\log d)^2 + d(\log d)^2|\logε|}$. At the present time (1993), this complexity is the best known in terms of the degree.