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Minimal models for rational functions in a dynamical setting

Part of: Algebraic number theory: local and $p$-adic fields Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  01 December 2012

Nils Bruin  and
Alexander Molnar
Nils Bruin
Affiliation:
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC V5A 1S6, Canada (email: nbruin@sfu.ca)
Alexander Molnar
Affiliation:
Mathematics and Statistics, Queen’s University, Jeffery Hall, University Avenue Kingston, ON K7L 3N6, Canada (email: a.molnar@queensu.ca)

Abstract

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We present a practical algorithm to compute models of rational functions with minimal resultant under conjugation by fractional linear transformations. We also report on a search for rational functions of degrees 2 and 3 with rational coefficients that have many integers in a single orbit. We find several minimal quadratic rational functions with eight integers in an orbit and several minimal cubic rational functions with ten integers in an orbit. We also make some elementary observations on possibilities of an analogue of Szpiro’s conjecture in a dynamical setting and on the structure of the set of minimal models for a given rational function.

MSC classification

Secondary: 37P05: Polynomial and rational maps 11S82: Non-Archimedean dynamical systems

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 400 - 417
Copyright
© The Author(s) 2012

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