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, $a,b\in ℂ\left[x\right]$, there is a polynomial $h$ such that for all $n$, we have

$$gcd\left(a^n-1,b^n-1\right)|h$$

We prove a compositional analog of this theorem, namely that if $f,g\in ℂ\left[x\right]$ are compositionally independent polynomials and $c\left(x\right)\in ℂ\left[x\right]$, then there are at most finitely many $\lambda$ with the property that there is an $n$ such that $\left(x-\lambda \right)$ divides $gcd\left(f^{\circ n}\left(x\right)-c\left(x\right),g^{\circ n}\left(x\right)-c\left(x\right)\right)$.