Index divisibility in the orbit of 0 for integral polynomials

TL;DR

This paper investigates the properties of the index divisibility set for iterates of integral polynomials, especially when the sequence is a rigid divisibility sequence, and characterizes cases with finite divisibility sets for specific polynomial forms.

Contribution

It generalizes previous results on index divisibility sets for rigid divisibility sequences and classifies when certain polynomial maps have finite index divisibility sets.

Findings

Properties of index divisibility sets for rigid divisibility sequences

Characterization of polynomials with finite index divisibility sets

Extension of previous results by Chen, Stange, and the first author

Abstract

Let $f(x) \in \bbz[x]$ and consider the index divisibility set $D = \{n \in \bbn : n \mid f^n(0)\}$. We present a number of properties of $D$ in the case that $(f^n(0))_{n=1}^\infty$ is a rigid divisibility sequence, generalizing a number of results of Chen, Stange, and the first author. We then study the polynomial $x^d + x^e + c \in \bbz[x]$, where $d > e \ge 2$ and determine all cases where this map has a finite index divisibility set.