Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems
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- by Carlos D’Andrea, Alina Ostafe, Igor E. Shparlinski and Martín Sombra PDF
- Trans. Amer. Math. Soc. 371 (2019), 1169-1198 Request permission
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.
References
Additional Information
- Carlos D’Andrea
- Affiliation: Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via 585, 08007 Barcelona, Spain
- MR Author ID: 652039
- Email: cdandrea@ub.edu
- Alina Ostafe
- Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
- MR Author ID: 884181
- Email: alina.ostafe@unsw.edu.au
- Igor E. Shparlinski
- Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
- MR Author ID: 192194
- 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
- MR Author ID: 621582
- Email: sombra@ub.edu
- 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 - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1169-1198
- MSC (2010): Primary 37P05; Secondary 11G25, 11G35, 13P15, 37P25
- DOI: https://doi.org/10.1090/tran/7437
- MathSciNet review: 3885175