{"paper":{"title":"Full Classification of permutation rational functions and complete rational functions of degree three over finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","cs.DM"],"primary_cat":"math.NT","authors_text":"Andrea Ferraguti, Giacomo Micheli","submitted_at":"2018-05-08T15:14:45Z","abstract_excerpt":"Let $q$ be a prime power, $\\mathbb F_q$ be the finite field of order $q$ and $\\mathbb F_q(x)$ be the field of rational functions over $\\mathbb F_q$. In this paper we classify all rational functions $\\varphi\\in \\mathbb F_q(x)$ of degree 3 that induce a permutation of $\\mathbb P^1(\\mathbb F_q)$. Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03097","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":""},"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"}