# On the factorization of iterated polynomials

### Lucas Reis

Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

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## Abstract

Let $\mathbb{F}_q$ be the finite field with $q$ elements and $f, g\in \mathbb{F}_q[x]$ be polynomials of degree at least one. This paper deals with the asymptotic growth of certain arithmetic functions associated to the factorization of the iterated polynomials $f(g^{(n)}(x))$ over $\mathbb{F}_q$, such as the largest degree of an irreducible factor and the number of irreducible factors. In particular, we provide significant improvements on the results of D. Gómez-Pérez, A. Ostafe and I. Shparlinski (2014).

## Cite this article

Lucas Reis, On the factorization of iterated polynomials. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 1957–1978

DOI 10.4171/RMI/1187