On the factorization of iterates of $x^d+c$ in large degree

TL;DR

This paper investigates the factorization properties of iterates of the polynomial $x^d + c$ over function and number fields, showing that for many parameters, these iterates have a bounded number of factors, with applications to prime divisors and integral points.

Contribution

It proves that for many degrees and parameters, the iterates of $x^d + c$ have at most $d$ factors, extending understanding of polynomial iteration in arithmetic dynamics.

Findings

For many $c$ and $d$, $f_{d,c}^n(x)-eta$ has at most $d$ factors for all $n",

The set of degrees $d$ with this property has positive asymptotic density

Applied results to density of prime divisors and finiteness of integral points in orbits

Abstract

Let $K$ be a function field of a curve in characteristic zero or a number field over which the $abc$-conjecture holds, fix $\alpha\in K$, and let $f_{d,c}(x)=x^d+c$ for some $d\geq2$ and some $c\in K$. Then for many $c$ and $d$, we prove that $f_{d,c}^n(x)-\alpha$ has at most $d$ factors in $K[x]$ for all $n\geq1$. For example, when $\alpha=0$ we prove that the set \[\Big\{d\,:\, f_{d,c}^n(x)\;\text{has at most $d$ factors in $K[x]$ for all $n\geq1$ and all $h(c)>0$}\Big\}\] has positive asymptotic density. We then apply this result to compute the density of prime divisors in certain forward orbits and to establish the finiteness of integral points in certain backward orbits.