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A new proof of Carlitz-Wan conjecture on exceptional polynomials

Yilong Hu, Zhiyao Zhang

Abstract

We give a new proof of Carlitz-Wan's conjecture, previously proved by Lenstra (1995). Compared to Lenstra's proof, our argument is easier to follow and more intuitive.

A new proof of Carlitz-Wan conjecture on exceptional polynomials

Abstract

We give a new proof of Carlitz-Wan's conjecture, previously proved by Lenstra (1995). Compared to Lenstra's proof, our argument is easier to follow and more intuitive.
Paper Structure (5 sections, 4 theorems, 7 equations)

This paper contains 5 sections, 4 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions of algebraic geometry
  4. Lower bound for $\#X(\mathbb{F}_{q})-q-1$
  5. Proof of Conjecture

Key Result

Theorem 1

Let $f \in \mathbb{F}_q[x]$ be a polynomial. Define the following binary polynomial over $\mathbb{F}_q$: Then $f$ is an exceptional polynomial over $\mathbb{F}_q$ if and only if every irreducible factor of $F(x,y)$ over $\mathbb{F}_q$ can be further decomposed in a finite extension of $\mathbb{F}_{q}$. In other words, the only absolutely irreducible factor of $f(x)-f(y)$ over $\mathbb{F}_q$ is $x

Theorems & Definitions (5)

  • Theorem 1: Cohen
  • Theorem 2: Carlitz-Wan
  • Theorem 3: Weil
  • Theorem 4: Bombieri & Katz bombieri2010note
  • Remark 1