{"paper":{"title":"A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.DS","authors_text":"B\\'ela Bollob\\'as, Dierk Schleicher, Malte Lackmann","submitted_at":"2010-09-09T18:31:59Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1843","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}