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

TL;DR

This paper solves the longstanding Davenport-Lewis-Schinzel problem on polynomial reducibility, providing explicit conditions for fiber reducibility and applications to arithmetic dynamics and functional equations.

Contribution

It offers an almost-complete solution to the DLS problem and applies it to stability of polynomial iterates and functional equations in complex rational functions.

Findings

Explicit description of integers with reducible fibers of polynomial maps

Solution to the DLS problem in the context of the Hilbert-Siegel problem

Applications to stability in arithmetic dynamics and functional equations

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 $X,Y\in\mathbb{C}(z)$.