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 $\gcd(f^{ \circ n}(x) - c(x), g^{ \circ n}(x) - c(x))$.
Categories: math.NT math.DS