Jung Kyu Canci^[1] and Laura Paladino^[2]

On preperiodic points of rational functions defined over \({\mathbb F}_p(t)\)

Pages: 193-203
Received: 31 December 2015
Accepted: 29 February 2016
Mathematics Subject Classification (2010): 37P05, 37P35.
Keywords: Preperiodic points, function fields.
Author address:
[1] : Universität Basel, Mathematisches Institut, Spiegelgasse 1, Basel, CH-4051, Switzerland
[2] : University of Pisa, Department of Mathematics, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy

Laura Paladino's work is supported by Istituto Nazionale di Alta Matematica trough Assegno di Ricerca Ing. G. Schirillo

Abstract: Let \(P\in\mathbb{P}_1(\mathbb{Q})\) be a periodic point for a monic polynomial with coefficients in \(\mathbb{Z}\). With elementary techniques one sees that the minimal periodicity of \(P\) is at most \(2\). Recently we proved a generalization of this fact to the set of all rational functions defined over \({\mathbb Q}\) with good reduction everywhere (i.e. at any finite place of \(\mathbb{Q}\)). The set of monic polynomials with coefficients in \(\mathbb{Z}\) can be characterized, up to conjugation by elements in PGL\(_2({\mathbb Z})\), as the set of all rational functions defined over \(\mathbb{Q}\) with a totally ramified fixed point in \(\mathbb{Q}\) and with good reduction everywhere. Let \(p\) be a prime number and let \({\mathbb F}_p\) be the field with \(p\) elements. In the present paper we consider rational functions defined over the rational global function field \({\mathbb F}_p(t)\) with good reduction at every finite place. We prove some bounds for the cardinality of orbits in \({\mathbb F}_p(t)\cup \{\infty\}\) for periodic and preperiodic points.

References

[1] R. Benedetto, Preperiodic points of polynomials over global fields, J. Reine Angew. Math. 608 (2007), 123-153. MR2339471
[2] J. K. Canci, Cycles for rational maps of good reduction outside a prescribed set, Monatsh. Math. 149 (2006), 265-287. MR2284648
[3] J. K. Canci, Finite orbits for rational functions, Indag. Math. (N.S.) 18 (2007), no. 2, 203-214. MR2352676
[4] J. K. Canci and L. Paladino, Preperiodic points for rational functions defined over a global field in terms of good reduction, Proc. Amer. Math. Soc., to appear. DOI: 10.1090/proc/13096.
[5] M. Hindry and J. H. Silverman, Diophantine geometry. An introduction, Grad. Texts in Math., 201, Springer-Verlag, New York 2000. MR1745599
[6] L. Merel, Borne pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), 437-449. MR1369424
[7] P. Morton and J. H. Silverman, Rational periodic points of rational functions, Internat. Math. Res. Notices 1994, no. 2, 97-110. MR1264933
[8] P. Morton and J. H. Silverman, Periodic points, multiplicities, and dynamical units, J. Reine Angew. Math. 461 (1995), 81-122. MR1324210
[9] W. Narkiewicz, Polynomial cycles in algebraic number fields, Colloq. Math. 58 (1989), 151-155. MR1028168
[10] M. Rosen, Number theory in function fields, Grad. Texts in Math., 210, Springer-Verlag, New York 2002. MR1876657
[11] J. H. Silverman, The arithmetic of dynamical system, Grad. Texts in Math., 241, Springer, New York 2007. MR2316407
[12] H. Stichtenoth, Algebraic function field and codes, 2nd edition, Grad. Texts in Math., 254, Springer-Verlag, Berlin 2009. MR2464941
[13] U. Zannier, Lecture notes on Diophantine analysis, Appunti. Sc. Norm. Super. Pisa (N. S.), 8, Edizioni della Normale, Pisa 2009. MR2517762
[14] M. Zieve, Cycle of Polynomial Mappings, Ph.D. Thesis, University of California, Berkeley 1996. MR2694837