#### Vol. 11, No. 6, 2017

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Greatest common divisors of iterates of polynomials

### Liang-Chung Hsia and Thomas J. Tucker

Vol. 11 (2017), No. 6, 1437–1459
##### 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

$gcd\left({a}^{n}-1,{b}^{n}-1\right)\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}h$

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

##### Keywords
gcd, composition, heights, equidstribution
Primary: 37P05
Secondary: 14G25