# On irreducible divisors of iterated polynomials

### Domingo Gómez-Pérez

Universidad de Cantabria, Santander, Spain### Alina Ostafe

University of New South Wales, Sydney, Australia### Igor E. Shparlinski

University of New South Wales, Sydney, Australia

## 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 $q$).

## Cite this article

Domingo Gómez-Pérez, Alina Ostafe, Igor E. Shparlinski, On irreducible divisors of iterated polynomials. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1123–1134

DOI 10.4171/RMI/809