Rational Functions with Small Value Set

Abstract

In connection with Galois Theory and Algebraic Curves, this paper investigates rational functions $h(x) = f(x)/g(x) \in \mathbb{F}_q(x)$ for which the value set $V_h = {\{h(α) | α \in \mathbb{F}_q \cup\{\infty\}}\}$ is relatively small. In particular, under certain circumstances, it proves that $h(x)$ having a small value set is equivalent to the field extension $\mathbb{F}_q(x)/\mathbb{F}_q(h(x))$ being Galois.