On the factorization of iterated polynomials

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).