Orbits of Polynomial Dynamical Systems Modulo Primes
Number Theory
2017-02-09 v1 Dynamical Systems
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 . 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.
Keywords
Cite
@article{arxiv.1702.01976,
title = {Orbits of Polynomial Dynamical Systems Modulo Primes},
author = {Mei-Chu Chang and Carlos D'Andrea and Alina Ostafe and Igor E. Shparlinski and Martin Sombra},
journal= {arXiv preprint arXiv:1702.01976},
year = {2017}
}