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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.6105","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-05-27T05:05:42Z","cross_cats_sorted":["math-ph","math.MP","math.PR"],"title_canon_sha256":"934802bf4fff887e534c4ef17fcfc7b8e1e76c27e735bbd8d49b33d84cb6b422","abstract_canon_sha256":"0e2cfe2a07a8df922551facd9d35e93fdc45cc63a02e9faef21cf2bf3541630f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:45.612687Z","signature_b64":"vetrQ6Lo5/pJpvs80HGrTEX3DXvTUcDBr/tGYE6azU/Fwk2nErD6G3B0jj3bqJiD8pIfbH8d5LcVDgQUZaGbDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"661d8e57253e617abe0f7afada9ae5ee4917614df781ad44a7011ab4d5447a08","last_reissued_at":"2026-05-18T01:23:45.611996Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:45.611996Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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