Rational functions with small value set

Abstract

Let q be a prime power, and let Fq be the finite field with q elements. In connection with Galois theory and algebraic curves, this paper investigates rational functions h(x)=f(x)/g(x)∈Fq(x) for which the value sets Vh={h(α)|α∈Fq∪{∞}} are relatively small. In particular, under certain circumstances, it proves that h(x) having a small value set is equivalent to the field extension Fq(x)/Fq(h(x)) being Galois.