Galois points and rational functions with small value sets

Abstract

This paper presents a connection between Galois points and rational functions with small value sets over a finite field. This paper proves that a defining polynomial of any plane curve admitting two Galois points is an irreducible factor of a polynomial obtained from the equality of two rational functions in one variable for each. Under the assumption that Galois groups of two Galois points generate their semidirect product, a recent result of Bartoli, Borges, and Quoos indicates that one of these rational functions over a finite field has a very small value set. This paper shows that when two Galois points are external, the defining polynomial is an irreducible factor of the difference of two polynomials in one variable. This connects the study of Galois points to that of polynomials with small value sets.