Greatest common divisors of iterates of polynomials

TL;DR

This paper extends classical results on the gcd of polynomial iterates from integers to complex polynomials, establishing finiteness results for common divisors under composition independence.

Contribution

It proves a compositional analog of a gcd theorem for polynomials, showing finiteness of certain common divisors for compositionally independent polynomials.

Findings

Finiteness of roots dividing gcd of iterates for compositionally independent polynomials

Extension of gcd results from integers to complex polynomial composition

New bounds on common divisors in polynomial iteration

Abstract

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant compositionally independent polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $\lambda$ with the property that there is an $n$ such that $(x - \lambda)$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$.