Iterates of post-critically finite polynomials of the form $\boldsymbol{x^d+c}$

TL;DR

This paper investigates the arithmetic properties of post-critically finite polynomials of the form x^d + c, focusing on their iterates' factorization over their fields of definition and implications for Galois representations.

Contribution

It provides a detailed factorization of iterates of these polynomials and proves new cases of a conjecture on abelian arboreal Galois representations.

Findings

Factorization of polynomial iterates over their fields of definition

New cases of a conjecture on abelian arboreal Galois representations

Enhanced understanding of arithmetic properties of post-critically finite polynomials

Abstract

Fix a prime number $d$. The post-critically finite polynomials of the form $f_{d,c} = x^d+c\in \mathbb{C}[x]$ play a fundamental role in polynomial dynamics. While many results are known in the complex dynamical setting, much less is understood about the arithmetic properties of these polynomials. In this paper, we describe the factorization of the iterates of post-critically finite polynomials $f_{d,c}$ over their fields of definition. As a consequence, we prove new cases of a conjecture of Andrews and Petsche on abelian arboreal Galois representations.