{"paper":{"title":"The Variance of the Number of Zeros for Complex Random Polynomials Spanned by OPUC","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Aaron M. Yeager","submitted_at":"2019-08-06T16:19:22Z","abstract_excerpt":"Let $\\{\\varphi_k\\}_{k=0}^\\infty $ be a sequence of orthonormal polynomials on the unit circle (OPUC) with respect to a probability measure $ \\mu $. We study the variance of the number of zeros of random linear combinations of the form $$ P_n(z)=\\sum_{k=0}^{n}\\eta_k\\varphi_k(z), $$ where $\\{\\eta_k\\}_{k=0}^n $ are complex-valued random variables. Under the assumption that the distribution for each $\\eta_k$ satisfies certain uniform bounds for the fractional and logarithmic moments, for the cases when $\\{\\varphi_k\\}$ are regular in the sense of Ullman-Stahl-Totik or are such that the measure of o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02234","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/1908.02234/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"}