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The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$

Angelot Behajaina, Joachim König, Danny Neftin

Abstract

We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of $f(X)-g(Y)\in\mathbb C[X,Y]$. This yields an almost-complete solution to the Hilbert--Siegel problem: For a polynomial map $f$ whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of $f$ are reducible. We further apply the solution to stability of iterates of $f$ in arithmetic dynamics, and to solving the functional equation $f(X)=g(Y)$ in $X,Y\in\mathbb{C}(z)$.

The Davenport-Lewis-Schinzel problem on the reducibility of $f(X)-g(Y)$

Abstract

We solve the problem of Davenport--Lewis--Schinzel (DLS), originating in the 1950s, regarding the reducibility of . This yields an almost-complete solution to the Hilbert--Siegel problem: For a polynomial map whose composition factors avoid only very specific low-degree polynomials, we explicitly describe over which integers the fibers of are reducible. We further apply the solution to stability of iterates of in arithmetic dynamics, and to solving the functional equation in .

Paper Structure

This paper contains 18 sections, 44 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On the proof
  3. Related work
  4. Basic setup and Preliminaries
  5. Basic setup
  6. Monodromy groups
  7. Composing maps
  8. Reducibility
  9. Subdirect powers
  10. Wreath products of affine groups
  11. Lower bounds on solvable monodromy groups of polynomials
  12. Diagonality of the kernel
  13. The solvable case of Theorem \ref{['thm:DLS-C']}
  14. Proofs of main results
  15. DLS over general fields
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f,g\in \mathbb C[X]$ be of degree $>1$. Then $f(X)-g(Y)$ is reducible in $\mathbb{C}[X,Y]$ if and only if one of the following holds for some $f_1,g_1,\mu\in \mathbb{C}[X]\setminus \mathbb{C}$ with $\deg(\mu)=1$:

Figures (1)

  • Figure 1: Diagram of relevant field extensions in the proof of Lemma \ref{['clm:exisx0']}

Theorems & Definitions (95)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • Lemma 2.5
  • ...and 85 more