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