{"paper":{"title":"Pairing of Zeros and Critical Points for Random Meromorphic Functions on Riemann Surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.CV","authors_text":"Boris Hanin","submitted_at":"2013-05-27T05:05:42Z","abstract_excerpt":"We prove that zeros and critical points of a random polynomial $p_N$ of degree $N$ in one complex variable appear in pairs. More precisely, if $p_N$ is conditioned to have $p_N(\\xi)=0$ for a fixed $\\xi \\in \\C\\backslash\\set{0},$ we prove that there is a unique critical point z in the annulus $N^{-1-\\ep}<\\abs{z-\\xi}< N^{-1+\\ep}}$ and no critical points closer to $\\xi$ with probability at least $1-O(N^{-3/2+3\\ep}).$ We also prove an analogous statement in the more general setting of random meromorphic functions on a closed Riemann surface."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6105","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"}