On irreducible divisors of iterated polynomials

Abstract

D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials $f\in {\mathbb{F}}_q[X]$ over the finite field of $q$ elements, where $q$ is an odd prime power, their iterates eventually become reducible polynomials over ${\mathbb{F}}_q$. Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of $f$. In particular, we prove that the $n$th iterate of $f$ has a square-free divisor of degree of order at least $n^{1+o(1)}$ as $n\to \infty$ (uniformly in $q$).