Scarcity of cycles for rational functions over a number field
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- by Jung Kyu Canci and Solomon Vishkautsan PDF
- Trans. Amer. Math. Soc. 371 (2019), 335-356 Request permission
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.References
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Additional Information
- Jung Kyu Canci
- Affiliation: Universität Basel, Mathematisches Institut, Spiegelgasse $1$, CH-$4051$ Basel, Switzerland
- MR Author ID: 803697
- Email: jungkyu.canci@unibas.ch
- Solomon Vishkautsan
- Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
- Email: wishcow@gmail.com
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 335-356
- MSC (2010): Primary 37P05, 37P35
- DOI: https://doi.org/10.1090/tran/7217
- MathSciNet review: 3885146