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Preprint

Mei-Chu Chang , Carlos D'Andrea , Alina Ostafe , Igor E. Shparlinski , Martin Sombra

2017-02-07

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 slighly 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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Mart�n Sombra, Luis Miguel Pardo, Teresa Krick (2001)

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Joseph Silverman (2007)

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Khoa Nguyen, Thomas Tucker, Dragos Ghioca (2014)

http://arxiv.org/licenses/nonexclusive-distrib/1.0/