{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:FT6643LCLLUBD36QOMYQXRRFHT","short_pith_number":"pith:FT6643LC","schema_version":"1.0","canonical_sha256":"2cfdee6d625ae811efd073310bc6253cf2132b30fd4570963ac63ce2260dd1f1","source":{"kind":"arxiv","id":"1601.04382","version":1},"attestation_state":"computed","paper":{"title":"Connections between discriminants and the root distribution of polynomials with rational generating function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Khang Tran","submitted_at":"2016-01-18T01:35:48Z","abstract_excerpt":"Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\\sum_{m=0}^{\\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$, $D(t,z)=1+B(z)t+A(z)t^{3}$ and $D(t,z)=1+B(z)t+A(z)t^{4}$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_{m}(z)$ lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the $q$-analogue of the discriminant, a concept introduced by Mourad Ismail."},"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":"1601.04382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-01-18T01:35:48Z","cross_cats_sorted":[],"title_canon_sha256":"82acf99733a62b36e3178c1e88166cdc8f623906dd1ed341d779ea577072d9a2","abstract_canon_sha256":"a91ccdaf578d3afbf03102bf9d86c4e311a9cd209a70328e4c2c8b64f85e1bfc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:45.210700Z","signature_b64":"RVtYBRmSmmX7lDgYOWkJcOSwiUYe+iOi05IJ+gAxakskMbIc/MPVKAE8ylFZbhBwaeBYx7OCEPqaBdeiOsuJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2cfdee6d625ae811efd073310bc6253cf2132b30fd4570963ac63ce2260dd1f1","last_reissued_at":"2026-05-18T01:22:45.210216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:45.210216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connections between discriminants and the root distribution of polynomials with rational generating function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Khang Tran","submitted_at":"2016-01-18T01:35:48Z","abstract_excerpt":"Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\\sum_{m=0}^{\\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$, $D(t,z)=1+B(z)t+A(z)t^{3}$ and $D(t,z)=1+B(z)t+A(z)t^{4}$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_{m}(z)$ lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the $q$-analogue of the discriminant, a concept introduced by Mourad Ismail."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04382","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":"1601.04382","created_at":"2026-05-18T01:22:45.210288+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.04382v1","created_at":"2026-05-18T01:22:45.210288+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04382","created_at":"2026-05-18T01:22:45.210288+00:00"},{"alias_kind":"pith_short_12","alias_value":"FT6643LCLLUB","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FT6643LCLLUBD36Q","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FT6643LC","created_at":"2026-05-18T12:30:15.759754+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT","json":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT.json","graph_json":"https://pith.science/api/pith-number/FT6643LCLLUBD36QOMYQXRRFHT/graph.json","events_json":"https://pith.science/api/pith-number/FT6643LCLLUBD36QOMYQXRRFHT/events.json","paper":"https://pith.science/paper/FT6643LC"},"agent_actions":{"view_html":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT","download_json":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT.json","view_paper":"https://pith.science/paper/FT6643LC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.04382&json=true","fetch_graph":"https://pith.science/api/pith-number/FT6643LCLLUBD36QOMYQXRRFHT/graph.json","fetch_events":"https://pith.science/api/pith-number/FT6643LCLLUBD36QOMYQXRRFHT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT/action/storage_attestation","attest_author":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT/action/author_attestation","sign_citation":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT/action/citation_signature","submit_replication":"https://pith.science/pith/FT6643LCLLUBD36QOMYQXRRFHT/action/replication_record"}},"created_at":"2026-05-18T01:22:45.210288+00:00","updated_at":"2026-05-18T01:22:45.210288+00:00"}