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

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 defin...

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Bibliographic Details
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
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Summary: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.
ISSN:2331-8422