Rational functions with small value set

Let q be a prime power, and let F-q 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) is an element of F-q(x) for which the value sets V-h = {h(alpha) vertical bar alpha is an element of F-q boolean OR {infinity}} are relatively small. In particular, under certain circumstances, it proves that h(x) having a small value set is equivalent to the field extension F-q(x)/F-q(h(x)) being Galois. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates rational functions with relatively small value sets in connection with Galois theory and algebraic curves, and proves under certain circumstances that having a small value set is equivalent to the field extension being Galois.