{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:IWQ2ZFO4TG7VEOMA4Z2KZROKVR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"05e5b5f86a16434c236b065ceb8dd7fbc358e595333c1fb7f08c32fd24465016","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2021-09-22T19:55:08Z","title_canon_sha256":"04ebbe03ed5d920872bd713f5787582da02b98789ef664288b85fa50148e2e8d"},"schema_version":"1.0","source":{"id":"2109.11006","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.11006","created_at":"2026-07-05T03:23:14Z"},{"alias_kind":"arxiv_version","alias_value":"2109.11006v2","created_at":"2026-07-05T03:23:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.11006","created_at":"2026-07-05T03:23:14Z"},{"alias_kind":"pith_short_12","alias_value":"IWQ2ZFO4TG7V","created_at":"2026-07-05T03:23:14Z"},{"alias_kind":"pith_short_16","alias_value":"IWQ2ZFO4TG7VEOMA","created_at":"2026-07-05T03:23:14Z"},{"alias_kind":"pith_short_8","alias_value":"IWQ2ZFO4","created_at":"2026-07-05T03:23:14Z"}],"graph_snapshots":[{"event_id":"sha256:1d3b3e9f48cf312a7f7eb1c00860a401d24261787fb4b28b9cb4ef768cb09bf9","target":"graph","created_at":"2026-07-05T03:23:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.11006/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Erd\\H{o}s and Tur\\'an proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erd\\H{o}s-Tur\\'an inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a cla","authors_text":"Jiuya Wang, Ruiwen Shu","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2021-09-22T19:55:08Z","title":"The Sharp Erd\\H{o}s-Tur\\'an Inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.11006","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d54c081a67b921b5a68b2f1dd203a8cfc58bd9af637a525261a1a2dfe8894465","target":"record","created_at":"2026-07-05T03:23:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"05e5b5f86a16434c236b065ceb8dd7fbc358e595333c1fb7f08c32fd24465016","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2021-09-22T19:55:08Z","title_canon_sha256":"04ebbe03ed5d920872bd713f5787582da02b98789ef664288b85fa50148e2e8d"},"schema_version":"1.0","source":{"id":"2109.11006","kind":"arxiv","version":2}},"canonical_sha256":"45a1ac95dc99bf523980e674acc5caac5032ef543dc8ff0183544d935167e5a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45a1ac95dc99bf523980e674acc5caac5032ef543dc8ff0183544d935167e5a3","first_computed_at":"2026-07-05T03:23:14.399752Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:23:14.399752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wB0XIugcjlHJNVbbTXcQ6RhT1h09jM543hOG70FJqVQPyknYGZqhBoXvYN3auvFwwJZOProt0yjm+ECP44hDDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:23:14.400127Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.11006","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d54c081a67b921b5a68b2f1dd203a8cfc58bd9af637a525261a1a2dfe8894465","sha256:1d3b3e9f48cf312a7f7eb1c00860a401d24261787fb4b28b9cb4ef768cb09bf9"],"state_sha256":"3b59cbf2b4a03200d4fa465c3af3c5a0e5552b5ec95838aa56a59f4a6a424d49"}