{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:G2U63UFCKGPDJ2QZG2BKB63DE3","short_pith_number":"pith:G2U63UFC","schema_version":"1.0","canonical_sha256":"36a9edd0a2519e34ea193682a0fb6326cce909a1007aeafb1c34bd8e90735bed","source":{"kind":"arxiv","id":"1507.06101","version":2},"attestation_state":"computed","paper":{"title":"On a family of Laurent polynomials generated by 2x2 matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Victor Katsnelson","submitted_at":"2015-07-22T08:43:53Z","abstract_excerpt":"To a $2\\times2$ matrix $G$ with complex entries, we relate the sequence of Laurent polynomial $L_n(z,G)=\\tr \\big(G\\big[\\begin{smallmatrix}z&0\\\\ 0&z^{-1}\\end{smallmatrix}\\big]G^{\\ast}\\big)^n$. It turns out that for each \\(n\\), the family $\\big\\{L_n(z,G)\\big\\}_G$, where $G$ runs over the set of all $2\\times2$ matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial $L_n(z,G)$ is expressed in terms of these parameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial $L_n(z,G)$ is described."},"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":"1507.06101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-22T08:43:53Z","cross_cats_sorted":[],"title_canon_sha256":"57d1c59642f817bdad5dc474bc68195f5f7c36392ca6b4d3420977dd4db7d321","abstract_canon_sha256":"f71802f46b59049679492ebec2cf5a58db0287ceafea2796c15abc5773ef4993"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:53.046967Z","signature_b64":"rIkSO79v2MkO99TLhrE5gSHkdFt7DI7RBXVHgY3QbywyP6cH4jaTWlJd7WgW3TVaTzwq3Oa48cGYdEosv7+7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36a9edd0a2519e34ea193682a0fb6326cce909a1007aeafb1c34bd8e90735bed","last_reissued_at":"2026-05-18T01:14:53.046247Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:53.046247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a family of Laurent polynomials generated by 2x2 matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Victor Katsnelson","submitted_at":"2015-07-22T08:43:53Z","abstract_excerpt":"To a $2\\times2$ matrix $G$ with complex entries, we relate the sequence of Laurent polynomial $L_n(z,G)=\\tr \\big(G\\big[\\begin{smallmatrix}z&0\\\\ 0&z^{-1}\\end{smallmatrix}\\big]G^{\\ast}\\big)^n$. It turns out that for each \\(n\\), the family $\\big\\{L_n(z,G)\\big\\}_G$, where $G$ runs over the set of all $2\\times2$ matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial $L_n(z,G)$ is expressed in terms of these parameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial $L_n(z,G)$ is described."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06101","kind":"arxiv","version":2},"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":"1507.06101","created_at":"2026-05-18T01:14:53.046367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.06101v2","created_at":"2026-05-18T01:14:53.046367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06101","created_at":"2026-05-18T01:14:53.046367+00:00"},{"alias_kind":"pith_short_12","alias_value":"G2U63UFCKGPD","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"G2U63UFCKGPDJ2QZ","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"G2U63UFC","created_at":"2026-05-18T12:29:22.688609+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/G2U63UFCKGPDJ2QZG2BKB63DE3","json":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3.json","graph_json":"https://pith.science/api/pith-number/G2U63UFCKGPDJ2QZG2BKB63DE3/graph.json","events_json":"https://pith.science/api/pith-number/G2U63UFCKGPDJ2QZG2BKB63DE3/events.json","paper":"https://pith.science/paper/G2U63UFC"},"agent_actions":{"view_html":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3","download_json":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3.json","view_paper":"https://pith.science/paper/G2U63UFC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.06101&json=true","fetch_graph":"https://pith.science/api/pith-number/G2U63UFCKGPDJ2QZG2BKB63DE3/graph.json","fetch_events":"https://pith.science/api/pith-number/G2U63UFCKGPDJ2QZG2BKB63DE3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3/action/storage_attestation","attest_author":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3/action/author_attestation","sign_citation":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3/action/citation_signature","submit_replication":"https://pith.science/pith/G2U63UFCKGPDJ2QZG2BKB63DE3/action/replication_record"}},"created_at":"2026-05-18T01:14:53.046367+00:00","updated_at":"2026-05-18T01:14:53.046367+00:00"}