{"paper":{"title":"On Zeroes of Random Polynomials and Applications to Unwinding","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Hau-Tieng Wu, Stefan Steinerberger","submitted_at":"2018-07-15T18:01:12Z","abstract_excerpt":"Let $\\mu$ be a probability measure in $\\mathbb{C}$ with a continuous and compactly supported density function, let $z_1, \\dots, z_n$ be independent random variables, $z_i \\sim \\mu$, and consider the random polynomial $$ p_n(z) = \\prod_{k=1}^{n}{(z - z_k)}.$$ We determine the asymptotic distribution of $\\left\\{z \\in \\mathbb{C}: p_n(z) = p_n(0)\\right\\}$. In particular, if $\\mu$ is radial around the origin, then those solutions are also distributed according to $\\mu$ as $n \\rightarrow \\infty$. Generally, the distribution of the solutions will reproduce parts of $\\mu$ and condense another part on "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05587","kind":"arxiv","version":3},"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"}