Iterates of Generic Polynomials and Generic Rational Functions

TL;DR

This paper extends Odoni's 1985 results on the Galois groups of iterated generic polynomials and rational functions from characteristic zero to positive characteristic, with applications in number theory density results.

Contribution

It generalizes Odoni's theorem to positive characteristic fields and to generic rational functions, broadening the scope of Galois group analysis for iterates.

Findings

Galois group of iterates is the wreath power of symmetric group in positive characteristic

Results enable new density theorems in number theory

Extension from polynomials to rational functions

Abstract

In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.