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Some results on the Flynn–Poonen–Schaefer conjecture

Part of: Arithmetic and non-Archimedean dynamical systems Complex dynamical systems General field theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  11 August 2021

Shalom Eliahou  and
Youssef Fares
Shalom Eliahou
Affiliation:
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d’Opale, UR 2597, F-62100 Calais, France, and CNRS, FR2037, France e-mail: [email protected]
Youssef Fares*
Affiliation:
LAMFA, CNRS-UMR 7352, Université de Picardie, 80039 Amiens, France, and CNRS, FR2037, France
*
e-mail: [email protected]
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Abstract

For $c \in \mathbb {Q}$ , consider the quadratic polynomial map $\varphi _c(z)=z^2-c$ . Flynn, Poonen, and Schaefer conjectured in 1997 that no rational cycle of $\varphi _c$ under iteration has length more than $3$ . Here, we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we show that if $\varphi _c$ admits a rational cycle of length $n \ge 3$ , then the denominator of c must be divisible by $16$ . We then provide an upper bound on the number of periodic rational points of $\varphi _c$ in terms of the number s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for $\varphi _c$ if $s \le 2$ , i.e., if the denominator of c has at most two distinct prime factors.

Keywords

Rational quadratic polynomialpolynomial iterationdiscrete dynamical systemperiodic points

MSC classification

Primary: 37P35: Arithmetic properties of periodic points
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 12E10: Special polynomials 11S05: Polynomials
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 598 - 611
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Copyright
© Canadian Mathematical Society 2021

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References

Adam, A. and Fares, Y., On two affine-like dynamical systems in a local field . J. Number Theory. 132(2012), 2892–2906.CrossRefGoogle Scholar
Benedetto, R., Preperiodic points of polynomials over global fields . J. Reine Angew. Math. 608(2007), 123–153.Google Scholar
Call, G. and Goldstine, S., Canonical heights on projective space . J. Number Theory. 63(1997), 211–243.CrossRefGoogle Scholar
Eliahou, S. and Fares, Y., Poonen’s conjecture and Ramsey numbers . Discrete Appl. Math. 209(2016), 102–106.10.1016/j.dam.2015.07.038CrossRefGoogle Scholar
Fares, Y., On the iteration over ℝ of rational quadratic polynomials. Preprint, 2020. https://hal.archives-ouvertes.fr/hal-02534351/document Google Scholar
Flynn, E. V., Poonen, B., and Schaefer, E. F., Cycles of quadratic polynomials and rational points on a genus-2 curve . Duke Math. J. 90(1997), 435–463.10.1215/S0012-7094-97-09011-6CrossRefGoogle Scholar
Morton, P., Arithmetic properties of periodic points of quadratic maps . Acta Arith. 62(1992), 343–372.10.4064/aa-62-4-343-372CrossRefGoogle Scholar
Narkiewicz, W., Cycle-lengths of a class of monic binomials . Funct. Approx. Comment. Math. 42(2010), 2, 163–168.10.7169/facm/1277811639CrossRefGoogle Scholar
Narkiewicz, W., On a class of monic binomials . Proc. Steklov Inst. Math. 280(2013), suppl. 2, S65–S70.10.1134/S0081543813030073CrossRefGoogle Scholar
Northcott, D., Periodic points on an algebraic variety . Ann. of Math. (2). 51(1950), 167–177.10.2307/1969504CrossRefGoogle Scholar
Pezda, T., Polynomial cycles in certain local domains . Acta Arith. 66(1994), 11–22.10.4064/aa-66-1-11-22CrossRefGoogle Scholar
Poonen, B., The classification of rational preperiodic points of quadratic polynomials over ℚ: a refined conjecture. Math. Z. 228(1998), 11–29.10.1007/PL00004405CrossRefGoogle Scholar
Silverman, J. H., The arithmetic of dynamical systems, Graduate Texts in Mathematics, 241, Springer, New York, 2007.CrossRefGoogle Scholar
Stoll, M., Rational 6-cycles under iteration of quadratic polynomials . LMS J. Comput. Math. 11(2008), 367–380.10.1112/S1461157000000644CrossRefGoogle Scholar
Walde, R. and Russo, P., Rational periodic points of the quadratic function Qc = x 2 + c. Amer. Math. Monthly. 101(1994), 318–331.Google Scholar
Zieve, M., Cycles of polynomial mappings. Ph.D. thesis, UC Berkeley, 1996.Google Scholar