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).

中文翻译：

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关于迭代多项式的因式分解

令$ \ mathbb {F} _q $是具有$ q $元素的有限域，并且$ f，\ mathbb {F} _q [x] $中的g \是度数至少为1的多项式。本文讨论了与$ \ mathbb {F} _q $上的迭代多项式$ f（g ^ {（n）}（x））$的因式分解相关的某些算术函数的渐近增长，例如最大程度的不可约因子和不可约因子的数量。特别是，我们对D.Gómez-Pérez，A。Ostafe和I. Shparlinski（2014）的结果进行了重大改进。