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

• Published in: arXiv.org 2016-01
• Main Authors: Jung Kyu Canci, Paladino, Laura
• Format: Article
• Language: English
• Subjects: Codes; Conjugation; Error analysis; Fixed points (mathematics); Mathematical analysis; Periodic variations; Polynomials; Rational functions; Reduction
• ISSN: 2331-8422

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.