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We show that for any $0 < \\epsilon <1$, the value of $p(1)$ is determined within relative error $\\epsilon$ by the coefficients $a_k$ with $k \\leq {c \\over \\sqrt{\\delta}} \\ln {n \\over \\epsilon \\sqrt{ \\delta}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < \\epsilon < 1$, the total number $M(G)=m_0(G)+m_1(G) + \\ldots $ of matchings is determined within relative error $\\epsil"},"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":"1806.07404","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-19T18:01:23Z","cross_cats_sorted":["cs.DS","math.CA"],"title_canon_sha256":"d94a0eb1e1e96e3f0dd430e09798eb52c463541ccdb317961dba76c46c9ff75d","abstract_canon_sha256":"841fabfe355aa0902eb1663069f2f212afd5175003f80ac5e98dbb9fe3444cad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:47.843796Z","signature_b64":"SdST83eAejP2vU5U9bxWRJegre1mLDXBM70ds54Qd5nPpuIJuO6SVZxWF5C6rRYpxYDSoM+yGEVz+3CCKF9rCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9259bc91e04eb4776cc667884f5233b3ba67426a6a95c59a9aec8b06974e35df","last_reissued_at":"2026-05-18T00:12:47.843270Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:47.843270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximating real-rooted and stable polynomials, with combinatorial applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CA"],"primary_cat":"math.CO","authors_text":"Alexander Barvinok","submitted_at":"2018-06-19T18:01:23Z","abstract_excerpt":"Let $p(x)=a_0 + a_1 x + \\ldots + a_n x^n$ be a polynomial with all roots real and satisfying $x \\leq -\\delta$ for some $0<\\delta <1$. We show that for any $0 < \\epsilon <1$, the value of $p(1)$ is determined within relative error $\\epsilon$ by the coefficients $a_k$ with $k \\leq {c \\over \\sqrt{\\delta}} \\ln {n \\over \\epsilon \\sqrt{ \\delta}}$ for some absolute constant $c > 0$. Consequently, if $m_k(G)$ is the number of matchings with $k$ edges in a graph $G$, then for any $0 < \\epsilon < 1$, the total number $M(G)=m_0(G)+m_1(G) + \\ldots $ of matchings is determined within relative error $\\epsil"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07404","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.07404","created_at":"2026-05-18T00:12:47.843357+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.07404v1","created_at":"2026-05-18T00:12:47.843357+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.07404","created_at":"2026-05-18T00:12:47.843357+00:00"},{"alias_kind":"pith_short_12","alias_value":"SJM3ZEPAJ22H","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"SJM3ZEPAJ22HO3GG","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"SJM3ZEPA","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.21089","citing_title":"A rigorous quasipolynomial-time classical algorithm for SYK thermal expectations","ref_index":5,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO","json":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO.json","graph_json":"https://pith.science/api/pith-number/SJM3ZEPAJ22HO3GGM6EE6URTWO/graph.json","events_json":"https://pith.science/api/pith-number/SJM3ZEPAJ22HO3GGM6EE6URTWO/events.json","paper":"https://pith.science/paper/SJM3ZEPA"},"agent_actions":{"view_html":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO","download_json":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO.json","view_paper":"https://pith.science/paper/SJM3ZEPA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.07404&json=true","fetch_graph":"https://pith.science/api/pith-number/SJM3ZEPAJ22HO3GGM6EE6URTWO/graph.json","fetch_events":"https://pith.science/api/pith-number/SJM3ZEPAJ22HO3GGM6EE6URTWO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO/action/storage_attestation","attest_author":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO/action/author_attestation","sign_citation":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO/action/citation_signature","submit_replication":"https://pith.science/pith/SJM3ZEPAJ22HO3GGM6EE6URTWO/action/replication_record"}},"created_at":"2026-05-18T00:12:47.843357+00:00","updated_at":"2026-05-18T00:12:47.843357+00:00"}