On the orbit of a post-critically finite polynomial of the form $x^d + c$

TL;DR

This paper investigates the arithmetic properties of critical orbits of post-critically finite polynomials of the form $x^d + c$, establishing new results on the irreducibility of associated polynomials and the irreducibility of iterates over $\,\mathbb{Q}(c)$.

Contribution

It proves the irreducibility of certain polynomials $G_d(m,n)$ for infinitely many pairs $(m,n)$ when $d$ is prime, and shows all iterates are irreducible over $\,\mathbb{Q}(c)$ under specific conditions.

Findings

Proved irreducibility of $G_d(m,n)$ for infinitely many $(m,n)$ when $d$ is prime.

Established all iterates are irreducible over $\,\mathbb{Q}(c)$ if $f_{c,d}$ has a fixed point in its post-critical orbit.

First known infinite families of $(m,n)$ with irreducible $G_d(m,n)$ polynomials.

Abstract

In this paper, we study the critical orbit of a post-critically finite polynomial of the form $f_{c,d}(x) = x^d+c \in \mathbb{C}[x]$. We discover that in many cases the orbit elements satisfy some strong arithmetic properties. It is well known that the $c$ values for which $f_{c,d}$ has tail size $m\geq 1$ and period $n$ are the roots of a polynomial $G_d(m,n) \in \mathbb{Z}[x]$, and the irreducibility or not of $G_d(m,n)$ has been a great mystery. As a consequence of our work, for any prime $d$, we establish the irreducibility of these $G_d(m,n)$ polynomials for infinitely many pairs $(m,n)$. These appear to be the first known such infinite families of $(m,n)$. We also prove that all the iterates of $f_{c,d}$ are irreducible over $\mathbb{Q}(c)$ if $d$ is a prime and $f_{c,d}$ has a fixed point in its post-critical orbit.