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On irreducible divisors of iterated polynomials. (English) Zbl 1339.11099
Summary: D. Gómez-Pérez et al. [Rev. Mat. Iberoam. 30, No. 2, 523–535 (2014; Zbl 1319.11089)] 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 \(q\)).

MSC:
11T06 Polynomials over finite fields
11L40 Estimates on character sums
11T24 Other character sums and Gauss sums
11P99 Additive number theory; partitions
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