{"paper":{"title":"Equidistribution of zeros of random polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CV","authors_text":"Igor Pritsker, Koushik Ramachandran","submitted_at":"2016-07-11T08:21:25Z","abstract_excerpt":"We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \\sum_{k=0}^n A_k B_k(z)$, where $\\{A_k\\}_{k=0}^{\\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\\{B_k\\}_{k=0}^{\\infty}$ are deterministic, and are selected from a standard basis such as Szeg\\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\\mathbb{E}[\\log^+|A_0|]<\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02855","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"}