{"paper":{"title":"On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Friedrich G\\\"otze","submitted_at":"2011-04-28T11:37:25Z","abstract_excerpt":"Consider a random polynomial $G_n(z)=\\xi_nz^n+...+\\xi_1z+\\xi_0$ with i.i.d. complex-valued coefficients. Suppose that the distribution of $\\log(1+\\log(1+|\\xi_0|))$ has a slowly varying tail. Then the distribution of the complex roots of $G_n$ concentrates in probability, as $n\\to\\infty$, to two centered circles and is uniform in the argument as $n\\to\\infty$. The radii of the circles are $|\\xi_0/\\xi_\\tau|^{1/\\tau}$ and $|\\xi_\\tau/\\xi_n|^{1/(n-\\tau)}$, where $\\xi_\\tau$ denotes the coefficient with the maximum modulus."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5360","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"}