Galois points and rational functions with small value sets

TL;DR

This paper explores the relationship between Galois points on plane curves and rational functions with small value sets over finite fields, establishing new connections and characterizations in algebraic geometry.

Contribution

It proves that the defining polynomial of curves with two Galois points relates to rational functions with small value sets, extending previous results under specific group assumptions.

Findings

The defining polynomial is an irreducible component of a relation of two rational functions.

When Galois points are external, the polynomial has separated variables.

Connects Galois point theory with polynomials having small value sets.

Abstract

This paper presents a connection between Galois points and rational functions over a finite field with small value sets. This paper proves that the defining polynomial of any plane curve admitting two Galois points is an irreducible component of a polynomial obtained as a relation of two rational functions. A recent result of Bartoli, Borges, and Quoos implies that one of these rational functions over a finite field has a very small value set, under the assumption that Galois groups of two Galois points generate the semidrect product. When two Galois points are external, this paper proves that the defining polynomial is an irreducible component of a polynomial with separated variables. This connects the study of Galois points to that of polynomials with small value sets.