D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials over the finite field of elements, where is an odd prime power, their iterates eventually become reducible polynomials over . Here we combine their method with some new ideas to derive finer results about the arithmetic structure of iterates of . In particular, we prove that the th iterate of has a square-free divisor of degree of order at least as (uniformly in ).
Cite this article
Domingo Gómez-Pérez, Alina Ostafe, Igor E. Shparlinski, On irreducible divisors of iterated polynomials. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1123–1134