{"paper":{"title":"A quadratic approximation to the Sendov radius near the unit circle","license":"","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Michael Miller","submitted_at":"2003-10-01T01:35:08Z","abstract_excerpt":"Define $S(n,\\beta)$ to be the set of complex polynomials of degree $n \\ge 2$ with all roots in the unit disk and at least one root at $\\beta$. For a polynomial $P$, define $|P|_\\beta$ to be the distance between $\\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\\beta)=\\sup \\{|P|_\\beta : P \\in S(n,\\beta) \\}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\\beta) \\le 1$.\n  In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\\beta) \\sim 1 + C_1 (1-|\\beta|) + C_2 ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0310004","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0310004/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}