Orbits of polynomial dynamical systems modulo primes

Chang, Mei-Chu; D’Andrea, Carlos; Ostafe, Alina; Shparlinski, Igor; Sombra, Martín

doi:10.1090/proc/13904

Orbits of polynomial dynamical systems modulo primes
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by Mei-Chu Chang, Carlos D’Andrea, Alina Ostafe, Igor E. Shparlinski and Martín Sombra

Proc. Amer. Math. Soc. 146 (2018), 2015-2025

DOI: https://doi.org/10.1090/proc/13904

Published electronically: December 26, 2017

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Abstract:

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb {C}$. Applying recent results of Baker and DeMarco (2011) and of Ghioca, Krieger, Nguyen and Ye (2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang (2015). As a by-product, we also slightly improve a result of Silverman (2008) and recover a result of Akbary and Ghioca (2009) as special extreme cases of our estimates.

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