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

TL;DR

This paper introduces a small, probabilistic set of starting points for Newton's method that efficiently finds all roots of complex polynomials of fixed degree with high probability, outperforming deterministic sets.

Contribution

It presents a probabilistic universal set of initial points of size $O(d(\log\log d)^2)$ that intersects all root basins for polynomials of degree $d$, improving over deterministic methods.

Findings

Sets of size $O(d(\log\log d)^2)$ intersect all root basins with high probability.

Probabilistic sets outperform deterministic sets in size for root-finding.

Efficient universal starting points for Newton's method across all roots of fixed-degree polynomials.

Abstract

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.