{"paper":{"title":"An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Annamaria Iezzi, Xiang-dong Hou","submitted_at":"2019-06-22T19:09:25Z","abstract_excerpt":"We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\\in\\Bbb F_q(X)\\setminus\\{0\\}$ be such that $q$ is sufficiently large relative to $\\text{deg}\\, f$ and $\\text{deg}\\, g$, $f(\\Bbb F_q)\\subset g(\\Bbb F_q\\cup\\{\\infty\\})$, and for ``most'' $a\\in\\Bbb F_q\\cup\\{\\infty\\}$, $|\\{x\\in \\Bbb F_q:g(x)=g(a)\\}|>(\\text{deg}\\, g)/2$. Then there exists $h(X)\\in\\Bbb F_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.09487","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"}