## Scarcity of cycles for rational functions over a number field

HTML articles powered by AMS MathViewer

- 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

- Robert L. Benedetto,
*Preperiodic points of polynomials over global fields*, J. Reine Angew. Math.**608**(2007), 123–153. MR**2339471**, DOI 10.1515/CRELLE.2007.055 - Enrico Bombieri and Walter Gubler,
*Heights in Diophantine geometry*, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR**2216774**, DOI 10.1017/CBO9780511542879 - F. Beukers and H. P. Schlickewei,
*The equation $x+y=1$ in finitely generated groups*, Acta Arith.**78**(1996), no. 2, 189–199. MR**1424539**, DOI 10.4064/aa-78-2-189-199 - Jung Kyu Canci,
*Cycles for rational maps of good reduction outside a prescribed set*, Monatsh. Math.**149**(2006), no. 4, 265–287. MR**2284648**, DOI 10.1007/s00605-006-0387-7 - Jung Kyu Canci,
*Finite orbits for rational functions*, Indag. Math. (N.S.)**18**(2007), no. 2, 203–214. MR**2352676**, DOI 10.1016/S0019-3577(07)80017-6 - Jung Kyu Canci and Laura Paladino,
*Preperiodic points for rational functions defined over a global field in terms of good reduction*, Proc. Amer. Math. Soc.**144**(2016), no. 12, 5141–5158. MR**3556260**, DOI 10.1090/proc/13096 - J.-H. Evertse, H. P. Schlickewei, and W. M. Schmidt,
*Linear equations in variables which lie in a multiplicative group*, Ann. of Math. (2)**155**(2002), no. 3, 807–836. MR**1923966**, DOI 10.2307/3062133 - J.-H. Evertse,
*On equations in $S$-units and the Thue-Mahler equation*, Invent. Math.**75**(1984), no. 3, 561–584. MR**735341**, DOI 10.1007/BF01388644 - Jan-Hendrik Evertse,
*The number of solutions of decomposable form equations*, Invent. Math.**122**(1995), no. 3, 559–601. MR**1359604**, DOI 10.1007/BF01231456 - M. I. Kargapolov and Ju. I. Merzljakov,
*Fundamentals of the theory of groups*, Graduate Texts in Mathematics, vol. 62, Springer-Verlag, New York-Berlin, 1979. Translated from the second Russian edition by Robert G. Burns. MR**551207** - Daniel A. Marcus,
*Number fields*, Universitext, Springer-Verlag, New York-Heidelberg, 1977. MR**0457396** - Patrick Morton and Joseph H. Silverman,
*Rational periodic points of rational functions*, Internat. Math. Res. Notices**2**(1994), 97–110. MR**1264933**, DOI 10.1155/S1073792894000127 - Patrick Morton and Joseph H. Silverman,
*Periodic points, multiplicities, and dynamical units*, J. Reine Angew. Math.**461**(1995), 81–122. MR**1324210**, DOI 10.1515/crll.1995.461.81 - W. Narkiewicz,
*Polynomial cycles in algebraic number fields*, Colloq. Math.**58**(1989), no. 1, 151–155. MR**1028168**, DOI 10.4064/cm-58-1-151-155 - Władysław Narkiewicz,
*Polynomial mappings*, Lecture Notes in Mathematics, vol. 1600, Springer-Verlag, Berlin, 1995. MR**1367962**, DOI 10.1007/BFb0076894 - D. G. Northcott,
*Periodic points on an algebraic variety*, Ann. of Math. (2)**51**(1950), 167–177. MR**34607**, DOI 10.2307/1969504 - Joseph H. Silverman,
*The arithmetic of dynamical systems*, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR**2316407**, DOI 10.1007/978-0-387-69904-2 - Sebastian Troncoso,
*Bounds for preperiodic points for maps with good reduction*, J. Number Theory**181**(2017), 51–72. MR**3689669**, DOI 10.1016/j.jnt.2017.05.026

## 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