{"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"}