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March 2020 On the orbit of a post-critically finite polynomial of the form $x^d+c$
Vefa Goksel
Funct. Approx. Comment. Math. 62(1): 95-104 (March 2020). DOI: 10.7169/facm/1799

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.

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Vefa Goksel. "On the orbit of a post-critically finite polynomial of the form $x^d+c$." Funct. Approx. Comment. Math. 62 (1) 95 - 104, March 2020. https://doi.org/10.7169/facm/1799

Information

Published: March 2020
First available in Project Euclid: 26 October 2019

zbMATH: 07225502
MathSciNet: MR4074390
Digital Object Identifier: 10.7169/facm/1799

Subjects:
Primary: 11R09, 37P15

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.62 • No. 1 • March 2020
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