Monodromy groups of polynomials of composition length 2

TL;DR

This paper investigates the monodromy groups of compositions of two indecomposable polynomials, establishing conditions for their largeness or explicit exceptions, with implications for polynomial arithmetic problems.

Contribution

It provides a classification of monodromy groups for polynomial compositions of length two, identifying when they are large or exceptional, aiding in solving longstanding problems.

Findings

Monodromy groups are either large or in an explicit list of exceptions.

Largeness results are crucial for analyzing compositions of more than two polynomials.

Main result is a key step in solving a long-standing open problem.

Abstract

We study the monodromy groups of compositions of two indecomposable polynomials. In particular, we show that such monodromy groups either fulfill a certain "largeness" property, or are in an explicit list of exceptions. Such largeness results are crucial for dealing with compositions of more than two polynomials, and consequently are expected to have a wide range of applications to problems concerning the arithmetic of polynomials. Concretely, our main result is a key ingredient in the solution of a long-standing open problem due to Davenport, Lewis and Schinzel, achieved in a companion paper.