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A census of quadratic post-critically finite rational functions defined over $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Q}$

Part of: Arithmetic and non-Archimedean dynamical systems Algebraic number theory: global fields Computational number theory

Published online by Cambridge University Press:  01 July 2014

David Lukas ,
Michelle Manes  and
Diane Yap
David Lukas
Affiliation:
University of Hawaii, Honolulu, HI 96822,USA email dlukas@hawaii.edu
Michelle Manes
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822,USA email mmanes@math.hawaii.edu
Diane Yap
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822,USA email dianey@gmail.com

Abstract

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We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$, up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$. We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$.

MSC classification

Secondary: 37P05: Polynomial and rational maps 37P15: Global ground fields 11R99: None of the above, but in this section 11Y99: None of the above, but in this section
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 314 - 329
Copyright
© The Author(s) 2014 

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