Abstract: D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials f?Fq[X]f?Fq[X] over the finite field of qq elements, where qq is an odd prime power, their iterates eventually become reducible polynomials over FqFq. Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of ff. In particular, we prove that the nnth iterate of ff has a square-free divisor of degree of order at least n1+o(1)n1+o(1) as n?8n?8 (uniformly in qq).

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