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

Authors: Mei-Chu Chang, Carlos D’Andrea, Alina Ostafe, Igor E. Shparlinski and Martín Sombra
Journal: Proc. Amer. Math. Soc. 146 (2018), 2015-2025
MSC (2010): Primary 37P05; Secondary 11G25, 11G35, 13P15, 37P25
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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Additional Information

Mei-Chu Chang
Affiliation: Department of Mathematics, University of California. Riverside, California 92521
Email: mcc@math.ucr.edu

Carlos D’Andrea
Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain
Email: cdandrea@ub.edu

Alina Ostafe
Affiliation: School of Mathematics and Statistics, University of New South Wales. Sydney, NSW 2052, Australia
Email: alina.ostafe@unsw.edu.au

Igor E. Shparlinski
Affiliation: School of Mathematics and Statistics, University of New South Wales. Sydney, NSW 2052, Australia
Email: igor.shparlinski@unsw.edu.au

Martín Sombra
Affiliation: ICREA. Passeig Lluís Companys 23, 08010 Barcelona, Spain–and–Departament de Mat- emàtiques i Informàtica, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain
Email: sombra@ub.edu

DOI: https://doi.org/10.1090/proc/13904
Keywords: Algebraic dynamical system, preperiodic point, orbit length, polynomial equations, resultant
Received by editor(s): February 27, 2017
Received by editor(s) in revised form: June 28, 2017
Published electronically: December 26, 2017
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2017 American Mathematical Society