{"paper":{"title":"Exceptional rational functions of degree 5 over finite fields: classification by monodromy, ramification, and the Riemann-Hurwitz formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiang Fan, Zhichao Tang","submitted_at":"2026-07-06T13:37:42Z","abstract_excerpt":"We complete the classification of exceptional rational functions of degree 5 over any finite field $\\mathbb{F}_{q}$ of characteristic different from 2 and 5, up to M\\\"obius equivalence. Our approach employs the arithmetic and geometric monodromy groups, combined with the ramification structure of the associated cover and the Riemann-Hurwitz formula. The cyclic monodromy case yields precisely monomials and R\\'edei functions, while the dihedral case is resolved by analysing its inertia groups and branch points; this leads to Dickson polynomials and a non-polynomial family arising from rational 5"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05075","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05075/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"}