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Transactions of the American Mathematical Society

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Scarcity of cycles for rational functions over a number field


Authors: Jung Kyu Canci and Solomon Vishkautsan
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37P05, 37P35
DOI: https://doi.org/10.1090/tran/7217
Published electronically: April 25, 2018
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Abstract: We provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function and is linear in the degree. More generally, we show that there exists an explicit uniform bound on the number of periodic points for any rational function in a given finitely generated semigroup (under composition) of rational functions of degree at least 2. We show that under stronger assumptions the dependence on the degree of the map in the bounds can be removed.


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Additional Information

Jung Kyu Canci
Affiliation: Universität Basel, Mathematisches Institut, Spiegelgasse $1$, CH-$4051$ Basel, Switzerland
Email: jungkyu.canci@unibas.ch

Solomon Vishkautsan
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: wishcow@gmail.com

DOI: https://doi.org/10.1090/tran/7217
Received by editor(s): July 2, 2016
Received by editor(s) in revised form: November 28, 2016, and February 20, 2017
Published electronically: April 25, 2018
Additional Notes: The second author was partially supported by the ERC-Grant “Diophantine Problems,” No. 267273, during the preparation of this article.
Article copyright: © Copyright 2018 American Mathematical Society

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