On the factorization of iterated polynomials

TL;DR

This paper studies the asymptotic behavior of polynomial factorizations over finite fields, focusing on the growth of irreducible factors in iterated polynomials, and improves previous results in this area.

Contribution

It provides new asymptotic estimates for the factorization properties of iterated polynomials over finite fields, advancing understanding of their irreducible components.

Findings

Improved bounds on the largest degree of irreducible factors.

Enhanced estimates for the number of irreducible factors.

Refined asymptotic growth rates for factorization functions.

Abstract

Let $\mathbb F_q$ be the finite field with $q$ elements, $f, g\in \mathbb F_q[x]$ 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. G\'{o}mez-P\'{e}rez, A. Ostafe and I. Shparlinski (2014).