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Transactions of the American Mathematical Society

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Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems


Authors: Carlos D’Andrea, Alina Ostafe, Igor E. Shparlinski and Martín Sombra
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37P05; Secondary 11G25, 11G35, 13P15, 37P25
DOI: https://doi.org/10.1090/tran/7437
Published electronically: May 3, 2018
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Abstract: We give bounds for the number and the size of the primes $ p$ such that a reduction modulo $ p$ of a system of multivariate polynomials over the integers with a finite number $ T$ of complex zeros does not have exactly $ T$ zeros over the algebraic closure of the field with $ p$ elements.

We apply these bounds to study the periodic points and the intersection of orbits of algebraic dynamical systems over finite fields. In particular, we establish some links between these problems and the uniform dynamical Mordell-Lang conjecture.


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Additional Information

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

DOI: https://doi.org/10.1090/tran/7437
Keywords: Modular reduction of systems of polynomials, arithmetic Nullstellensatz, algebraic dynamical system, orbit length, orbit intersection
Received by editor(s): November 1, 2015
Received by editor(s) in revised form: April 27, 2017
Published electronically: May 3, 2018
Additional Notes: The first author was partially supported by the Spanish MEC research project MTM2013-40775-P
The second author was supported by the UNSW Vice Chancellor’s Fellowship
The third author was supported by the Australian Research Council Grants DP140100118 and DP170100786
The fourth author was supported by the Spanish MINECO research projects MTM2012-38122-C03-02 and MTM2015-65361-P
Article copyright: © Copyright 2018 American Mathematical Society

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