Hostname: page-component-cb9f654ff-lqqdg Total loading time: 0 Render date: 2025-08-25T09:30:06.964Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal models for rational functions in a dynamical setting

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

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Behnel, S., Bradshaw, R., Ewing, G., Seljebotn, D. S.et al., ‘Cython: C-extensions for python’, 2009, http://www.cython.org.Google Scholar
[2]Bosma, J., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.CrossRefGoogle Scholar
[3]Bruin, N. and Molnar, A., ‘Integers in orbits of rational functions ’, 2012, http://www.cecm.sfu.ca/∼nbruin/intorbits.Google Scholar
[4]Lang, S., Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 231 (Springer, Berlin, 1978); MR 518817(81b:10009).CrossRefGoogle Scholar
[5]Mahler, K., ‘On the lattice points on curves of genus 1’, Proc. Lond. Math. Soc., II. Ser. 39 (1935) 431–466; doi:10.1112/plms/s2-39.1.431.CrossRefGoogle Scholar
[6]Molnar, A., ‘Fractional linear minimal models of rational functions’, MSc Thesis, Simon Fraser University, 2011.Google Scholar
[7]Morton, P. and Silverman, J. H., ‘Rational periodic points of rational functions’, Int. Math. Res. Not. 2 (1994) 97–110; MR 1264933(95b:11066), doi:10.1155/S1073792894000127.CrossRefGoogle Scholar
[8]Silverman, J. H., ‘Integer points, Diophantine approximation, and iteration of rational maps’, Duke Math. J. 71 (1993) no. 3, 793–829; MR 1240603(95e:11070), doi:10.1215/S0012-7094-93-07129-3.CrossRefGoogle Scholar
[9]Silverman, J. H., ‘The field of definition for dynamical systems on P1’, Compositio Math. 98 (1995) no. 3, 269–304; MR 1351830(96j:11090).Google Scholar
[10]Silverman, J. H., The arithmetic of dynamical systems, Graduate Texts in Mathematics 241 (Springer, New York, 2007), MR 2316407(2008c:11002).CrossRefGoogle Scholar
[11]Stein, W. A., ‘Sage Mathematics Software (Version 4.7)’, The Sage Development Team, 2011, http://www.sagemath.org.Google Scholar
[12]Szpiro, L., Tepper, M. and Williams, P., ‘Resultant and conductor of geometrically semi-stable self maps of the projective line over a number field or function field’, Preprint, 2010, http://arxiv.org/abs/1010.5030.Google Scholar
[13]Tate, J., ‘Algorithm for determining the type of a singular fiber in an elliptic pencil’, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Mathematics 476 (Springer, Berlin, 1975) 33–52; MR 0393039(52#13850).Google Scholar