On the dynamical Galois group of certain affine polynomials in positive characteristic

TL;DR

This paper establishes a criterion using class field theory to determine when certain affine polynomials over fields of positive characteristic have maximal dynamical Galois groups, extending known results to characteristic 2.

Contribution

It introduces a new dynamical Galois criterion for polynomials of the form x^{p^r}+ax+b in positive characteristic, generalizing previous quadratic cases.

Findings

Provides explicit criterion for maximal dynamical Galois groups

Extends Stoll's quadratic polynomial criterion to characteristic 2

Connects dynamical Galois groups with class field theory in positive characteristic

Abstract

We use explicit class field theory of rational function fields to prove a dynamical criterion for a polynomial of the form $x^{p^r}+ax+b$ over a field of characteristic $p$ to have dynamical Galois group as large as possible. When $p=2$ and $r=1$ this yields an analogue in characteristic $2$ of the celebrated criterion of Stoll for quadratic polynomials over fields of characteristic not $2$.