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On a Class of Permutation Rational Functions Involving Trace Maps

Ruikai Chen, Sihem Mesnager

Abstract

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates.

On a Class of Permutation Rational Functions Involving Trace Maps

Abstract

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on extensions of finite fields, especially for the cases of quadratic and cubic extensions. Our achievements are obtained by investigating absolute irreducibility of some polynomials in two indeterminates.
Paper Structure (4 sections, 7 theorems, 101 equations)

This paper contains 4 sections, 7 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. The Case That $L$ Does Not Permute $\mathbb F_{q^n}$
  3. The Case That $L$ Permutes $\mathbb F_{q^n}$
  4. Conclusions

Key Result

Lemma 1

For an absolutely irreducible affine plane curve of degree $d$ over $\mathbb F_q$ with $d>1$, if then it has a rational point $(x_0,y_0)$ over $\mathbb F_q$ with $x_0\ne y_0$.

Theorems & Definitions (15)

  • Lemma 1: aubry1996weil
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Remark 6
  • ...and 5 more