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Galois points and rational functions with small value sets

Satoru Fukasawa

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.

Galois points and rational functions with 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.

Paper Structure

This paper contains 2 sections, 3 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem \oldthetheorem

Let $C \subset \mathbb{P}^2$ be defined over a finite field $\mathbb{F}_q$ of $q$ elements. Assume that $C$ is irreducible over the algebraic closure $\overline{\mathbb{F}}_q$ of $\mathbb{F}_q$. Let $P_1=(1:0:0), P_2=(0:1:0) \in \mathbb{P}^2$. If $P_1$ and $P_2$ are Galois points such that all autom Hereafter, we assume the existence of polynomials $f_1, g_1, f_2, g_2 \in \mathbb{F}_q[x]$ satisfyi

Theorems & Definitions (7)

  • Theorem \oldthetheorem
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Remark 2
  • proof : Proof of Theorem \ref{['main']}
  • proof : Proof of Corollary \ref{['Frobenius nonclassical']}