Greatest common divisors of iterates of polynomials

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 $\gcd (f^{\circ n}(x)-c(x),g^{\circ n}(x)-c(x))$.

22 pages