# Revista Matemática Iberoamericana

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**Volume 30, Issue 4, 2014, pp. 1123–1134**

**DOI: 10.4171/RMI/809**

Published online: 2014-12-15

On irreducible divisors of iterated polynomials

Domingo Gómez-Pérez^{[1]}, Alina Ostafe

^{[2]}and Igor E. Shparlinski

^{[3]}(1) Universidad de Cantabria, Santander, Spain

(2) University of New South Wales, Sydney, Australia

(3) University of New South Wales, Sydney, Australia

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 $q$).

*Keywords: *iterations of polynomials, irreducible divisors

Gómez-Pérez Domingo, Ostafe Alina, Shparlinski Igor: On irreducible divisors of iterated polynomials. *Rev. Mat. Iberoamericana* 30 (2014), 1123-1134. doi: 10.4171/RMI/809