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We consider a function I(x,y)=\\exp \\{h_2(x) A_1(x,y) \\diagup A_0(x,y) \\} h_1(x) \\prod_{i=1}^{\\ell} (y-g_i(x))^{\\alpha_i}, where g_i(x) are algebraic functions, A_1(x,y)=\\prod_{k=1}^r (y-a_k(x)), A_0(x,y)=\\prod_{j=1}^s (y-\\tilde{g}_j(x)) with a_k(x) and \\tilde{g}_j(x) algebraic functions, A_0 and A_1 do not share any common factor, h_2(x) is a rational function, h(x) and h_1(x) are functions with a rational logarithmic derivative and \\alpha_i are complex numbers. We show that if I(x,y) is a first integral"},"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":"math/0506036","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2005-06-02T08:28:11Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"299cbb95d9263c93c36b47296d9a91e6e2a8a6df5e8be1850d7afbfbc586e15f","abstract_canon_sha256":"eca16937d2e5a30587d0dfbd8395b7b091df604eeb70a15d0705790f164c7f28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:18.919793Z","signature_b64":"xC5jeY0iqA00U6x3AdiU3aDA/d/q1PPS8KxbOTgahVgk8RtymirILRYb26/S6u5LuX4NaxgMs9fz1GirlJ/6DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73ca58601adff5362c44ab1602fa8f79d284fec0544b27d6a2f201b37bf25bac","last_reissued_at":"2026-05-18T00:44:18.919261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:18.919261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The role of algebraic solutions in planar polynomial differential systems","license":"","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DS","authors_text":"H\\'ector Giacomini, Jaume Gin\\'e, Maite Grau","submitted_at":"2005-06-02T08:28:11Z","abstract_excerpt":"We study a planar polynomial differential system, given by \\dot{x}=P(x,y), \\dot{y}=Q(x,y). We consider a function I(x,y)=\\exp \\{h_2(x) A_1(x,y) \\diagup A_0(x,y) \\} h_1(x) \\prod_{i=1}^{\\ell} (y-g_i(x))^{\\alpha_i}, where g_i(x) are algebraic functions, A_1(x,y)=\\prod_{k=1}^r (y-a_k(x)), A_0(x,y)=\\prod_{j=1}^s (y-\\tilde{g}_j(x)) with a_k(x) and \\tilde{g}_j(x) algebraic functions, A_0 and A_1 do not share any common factor, h_2(x) is a rational function, h(x) and h_1(x) are functions with a rational logarithmic derivative and \\alpha_i are complex numbers. We show that if I(x,y) is a first integral"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0506036","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":"math/0506036","created_at":"2026-05-18T00:44:18.919346+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0506036v1","created_at":"2026-05-18T00:44:18.919346+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0506036","created_at":"2026-05-18T00:44:18.919346+00:00"},{"alias_kind":"pith_short_12","alias_value":"OPFFQYA2372T","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"OPFFQYA2372TMLCE","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"OPFFQYA2","created_at":"2026-05-18T12:25:53.335082+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/OPFFQYA2372TMLCEVMLAF6UPPH","json":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH.json","graph_json":"https://pith.science/api/pith-number/OPFFQYA2372TMLCEVMLAF6UPPH/graph.json","events_json":"https://pith.science/api/pith-number/OPFFQYA2372TMLCEVMLAF6UPPH/events.json","paper":"https://pith.science/paper/OPFFQYA2"},"agent_actions":{"view_html":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH","download_json":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH.json","view_paper":"https://pith.science/paper/OPFFQYA2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0506036&json=true","fetch_graph":"https://pith.science/api/pith-number/OPFFQYA2372TMLCEVMLAF6UPPH/graph.json","fetch_events":"https://pith.science/api/pith-number/OPFFQYA2372TMLCEVMLAF6UPPH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH/action/storage_attestation","attest_author":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH/action/author_attestation","sign_citation":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH/action/citation_signature","submit_replication":"https://pith.science/pith/OPFFQYA2372TMLCEVMLAF6UPPH/action/replication_record"}},"created_at":"2026-05-18T00:44:18.919346+00:00","updated_at":"2026-05-18T00:44:18.919346+00:00"}