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
MathSciNet review:
3767353
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Abstract | References | Similar Articles | Additional Information
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.
- Amir Akbary and Dragos Ghioca, Periods of orbits modulo primes, J. Number Theory 129 (2009), no. 11, 2831–2842. MR 2549537, DOI https://doi.org/10.1016/j.jnt.2009.03.007
- Vladimir Anashin and Andrei Khrennikov, Applied algebraic dynamics, De Gruyter Expositions in Mathematics, vol. 49, Walter de Gruyter & Co., Berlin, 2009. MR 2533085
- Matthew Baker and Laura DeMarco, Preperiodic points and unlikely intersections, Duke Math. J. 159 (2011), no. 1, 1–29. MR 2817647, DOI https://doi.org/10.1215/00127094-1384773
- Matthew Baker and Laura De Marco, Special curves and postcritically finite polynomials, Forum Math. Pi 1 (2013), e3, 35. MR 3141413, DOI https://doi.org/10.1017/fmp.2013.2
- Robert L. Benedetto, Dragos Ghioca, Benjamin Hutz, Pär Kurlberg, Thomas Scanlon, and Thomas J. Tucker, Periods of rational maps modulo primes, Math. Ann. 355 (2013), no. 2, 637–660. MR 3010142, DOI https://doi.org/10.1007/s00208-012-0799-8
- Mei-Chu Chang, On periods modulo $p$ in arithmetic dynamics, C. R. Math. Acad. Sci. Paris 353 (2015), no. 4, 283–285 (English, with English and French summaries). MR 3319121, DOI https://doi.org/10.1016/j.crma.2015.01.007
- Carlos D’Andrea, Teresa Krick, and Martín Sombra, Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 4, 549–627 (2013) (English, with English and French summaries). MR 3098424, DOI https://doi.org/10.24033/asens.2196
- C. D’Andrea, A. Ostafe, I. E. Shparlinski and M. Sombra, ‘Reductions modulo primes of systems of polynomial equations and algebraic dynamical systems’, Preprint, 2015 (see http://arxiv.org/1505.05814) to appear in Transactions of the American Mathematical Society, https://doi.org/10.1090/tran/7437
- Domingo Gomez, Jaime Gutierrez, Álvar Ibeas, and David Sevilla, Common factors of resultants modulo $p$, Bull. Aust. Math. Soc. 79 (2009), no. 2, 299–302. MR 2496933, DOI https://doi.org/10.1017/S0004972708001275
- Dragos Ghioca, Liang-Chung Hsia, and Thomas J. Tucker, Preperiodic points for families of polynomials, Algebra Number Theory 7 (2013), no. 3, 701–732. MR 3095224, DOI https://doi.org/10.2140/ant.2013.7.701
- D. Ghioca, L.-C. Hsia, and T. J. Tucker, Preperiodic points for families of rational maps, Proc. Lond. Math. Soc. (3) 110 (2015), no. 2, 395–427. MR 3335283, DOI https://doi.org/10.1112/plms/pdu051
- Dragos Ghioca, Holly Krieger, and Khoa Nguyen, A case of the dynamical André-Oort conjecture, Int. Math. Res. Not. IMRN 3 (2016), 738–758. MR 3493432, DOI https://doi.org/10.1093/imrn/rnv143
- D. Ghioca, H. Krieger, K. D. Nguyen, and H. Ye, The dynamical André-Oort conjecture: unicritical polynomials, Duke Math. J. 166 (2017), no. 1, 1–25. MR 3592687, DOI https://doi.org/10.1215/00127094-3673996
- Dragos Ghioca, Khoa Nguyen, and Thomas J. Tucker, Portraits of preperiodic points for rational maps, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 1, 165–186. MR 3349337, DOI https://doi.org/10.1017/S0305004115000274
- Patrick Ingram, A finiteness result for post-critically finite polynomials, Int. Math. Res. Not. IMRN 3 (2012), 524–543. MR 2885981, DOI https://doi.org/10.1093/imrn/rnr030
- Sergei V. Konyagin and Igor E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Tracts in Mathematics, vol. 136, Cambridge University Press, Cambridge, 1999. MR 1725241
- Teresa Krick, Luis Miguel Pardo, and Martín Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J. 109 (2001), no. 3, 521–598. MR 1853355, DOI https://doi.org/10.1215/S0012-7094-01-10934-4
- Klaus Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995. MR 1345152
- Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407
- Joseph H. Silverman, Variation of periods modulo $p$ in arithmetic dynamics, New York J. Math. 14 (2008), 601–616. MR 2448661
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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
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–and–Departament de Mat- emàtiques i Informàtica, Universitat de Barcelona. Gran Via 585, 08007 Barcelona, Spain
MR Author ID:
621582
Email:
sombra@ub.edu
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
