An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields

Abstract

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 $f(X),g(X)\in {\mathbb{F}}_q(X)\setminus \{0\}$ be such that $q$ is sufficiently large relative to $\text{deg}f$ and $\text{deg}g$, $f({\mathbb{F}}_q)\subset g({\mathbb{F}}_q\cup \{\infty \})$, and for ``most'' $a\in {\mathbb{F}}_q\cup \{\infty \}$, $|\{x\in {\mathbb{F}}_q:g(x)=g(a)\}|>(\text{deg}g)/2$. Then there exists $h(X)\in {\mathbb{F}}_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included.

8 pages