JournalsrmiVol. 30, No. 4pp. 1123–1134

On irreducible divisors of iterated polynomials

  • Domingo Gómez-Pérez

    Universidad de Cantabria, Santander, Spain
  • Alina Ostafe

    University of New South Wales, Sydney, Australia
  • Igor E. Shparlinski

    University of New South Wales, Sydney, Australia
On irreducible divisors of iterated polynomials cover

Abstract

D. Gómez-Pérez, A. Ostafe, A.P. Nicolás and D. Sadornil have recently shown that for almost all polynomials fFq[X]f \in \mathbb F_q[X] over the finite field of qq elements, where qq is an odd prime power, their iterates eventually become reducible polynomials over Fq\mathbb F_q. 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)n^{1+o(1)} as nn\to \infty (uniformly in qq).

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

DOI 10.4171/RMI/809