Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

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

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

Pages: 193-203
Accepted: 29 February 2016
Mathematics Subject Classification (2010): 37P05, 37P35.
Keywords: Preperiodic points, function fields.
[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.

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