{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LFZ63GVARF3EHOXFPR5ZWTKCGN","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":"aaf79c43658526e8e675dbc0da38bc8a19bbf0070759b5dbe9ef702c1287a033","cross_cats_sorted":["math.CV","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T05:48:43Z","title_canon_sha256":"cbe476b18b600144c95602a5e4a22e0996c91df668529a12204d8856b02cc532"},"schema_version":"1.0","source":{"id":"1307.5594","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5594","created_at":"2026-05-18T00:40:42Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5594v2","created_at":"2026-05-18T00:40:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5594","created_at":"2026-05-18T00:40:42Z"},{"alias_kind":"pith_short_12","alias_value":"LFZ63GVARF3E","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LFZ63GVARF3EHOXF","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LFZ63GVA","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:d7db169d3f11aa055133fd7d6ba3e453bf3b2596b836199a5394dea5ae5571b0","target":"graph","created_at":"2026-05-18T00:40:42Z","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"},"paper":{"abstract_excerpt":"Let $\\mathbb R_t[\\theta]$ be the ring generated over $\\mathbb R$ by $\\cos\\theta$ and $\\sin\\theta$, and $\\mathbb R_t(\\theta)$ be its quotient field. In this paper we study the ways in which an element p of $\\mathbb R_t[\\theta]$ can be decomposed into a composition of functions of the form $p=R(q),$ where $\\mathbb R\\in \\mathbb R(x)$ and $q\\in \\mathbb R_t(\\theta)$. In particular, we describe all possible solutions of the functional equation $R_1(q_1)=R_2(q_2)$, where $R_1, R_2 \\in \\mathbb R[x]$ and $q_1,q_2\\in \\mathbb R_t[\\theta].$","authors_text":"F. Pakovich","cross_cats":["math.CV","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T05:48:43Z","title":"On decompositions of trigonometric polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5594","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":"sha256:7f46c05b3bc14ce49ce3551d0b3ef7be90e7f0b86cad107c3422d97326ebb3e2","target":"record","created_at":"2026-05-18T00:40:42Z","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":"aaf79c43658526e8e675dbc0da38bc8a19bbf0070759b5dbe9ef702c1287a033","cross_cats_sorted":["math.CV","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-22T05:48:43Z","title_canon_sha256":"cbe476b18b600144c95602a5e4a22e0996c91df668529a12204d8856b02cc532"},"schema_version":"1.0","source":{"id":"1307.5594","kind":"arxiv","version":2}},"canonical_sha256":"5973ed9aa0897643bae57c7b9b4d42334d8e21777e42c63da552470056cb773d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5973ed9aa0897643bae57c7b9b4d42334d8e21777e42c63da552470056cb773d","first_computed_at":"2026-05-18T00:40:42.203736Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:42.203736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bEK7s89pI7JZTwddrOfnYiDYBGvZ4BENmXkET0NCPB/F/R8Ix0C9rcTlT1gW1n7LKHhlOjR2EnwrxcOsJUxpDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:42.204453Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5594","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7f46c05b3bc14ce49ce3551d0b3ef7be90e7f0b86cad107c3422d97326ebb3e2","sha256:d7db169d3f11aa055133fd7d6ba3e453bf3b2596b836199a5394dea5ae5571b0"],"state_sha256":"bebe4521a779e2f5cd2b2faf680b98a875d7696c6b741d2216d85ad791b73d5b"}