{"paper":{"title":"A new approach to the Poincar\\'e center problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Isaac A. Garc\\'ia, Jaume Gin\\'e","submitted_at":"2026-03-10T17:32:40Z","abstract_excerpt":"We address the classical (degenerate or non-degenerate) center problem posed by Poincar\\'e in the 19th century for monodromic singularities of analytic families of planar vector fields $\\mathcal{X}$. We prove that every analytic center admits a Laurent inverse integrating factor $V$ in weighted polar coordinates. Moreover, we show that when $\\mathcal{X}$ has no local curves of zero angular speed, the Poincar\\'e map is analytic. Based on this result, we derive a theoretical procedure to determine parameter constraints within the family that characterize centers without curves of zero angular sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.09941","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2603.09941/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}