{"paper":{"title":"Estimating the Koebe radius for polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alex Stokolos, Andrey Smorodin, Dmitriy Dmitrishin","submitted_at":"2018-05-17T19:00:16Z","abstract_excerpt":"For a pair of conjugate trigonometrical polynomials $C (t) = \\sum_ { j = 1 } ^N { { a_j}\\cos jt }, S(t) = \\sum_ { j = 1 } ^N { { a_j}\\sin jt }$ with real coefficients and normalization ${a_1} = 1 $ we solve the extremal problem \\[\n  \\sup_ {a_2,...,a_N} \\left ({ \\min_t \\left\\{ {\\Re \\left ({ F\\left ({ { e^ {it} } } \\right) } \\right): \\Im \\left ({ F\\left ({ { e^ {it} } } \\right) } \\right) = 0 } \\right\\} } \\right) = -\\frac14 \\sec ^2\\frac\\pi{N + 2}. \\] We show that the solution is unique and is given by \\[ a_j^ {(0)} = \\frac {1} { { { U'_N}\\left ({\\cos \\frac{\\pi } { { N + 2 } } } \\right) } } { U' _"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06927","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":""},"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"}