On irreducible divisors of iterated polynomials

• Autores: Domingo Gómez Pérez , Alina Ostafe, Igor E. Shparlinski
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 30, Nº 4, 2014, págs. 1123-1134
• Idioma: inglés
• DOI: 10.4171/RMI/809
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• Resumen

  • D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials f∈Fq[X] over the finite field of q elements, where q is an odd prime power, their iterates eventually become reducible polynomials over Fq. 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 nth iterate of f has a square-free divisor of degree of order at least n1+o(1) as n→∞ (uniformly in q).