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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Preperiodic portraits for unicritical polynomials over a rational function field
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by John R. Doyle PDF
Trans. Amer. Math. Soc. 370 (2018), 3265-3288 Request permission

Abstract:

Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal {K} := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \mathcal {K}[z]$, and we ask the following question: If we fix $\alpha \in \mathcal {K}$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\mathfrak {p} \in \mathrm {Spec} K[t]$ such that, modulo $\mathfrak {p}$, the point $\alpha$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the answer is generally affirmative and explicitly giving all counterexamples. This extends previous work by the author in the case that $\alpha$ is a constant point.
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Additional Information
  • John R. Doyle
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • Address at time of publication: Department of Mathematics and Statistics, GTMH 330, Louisiana Tech University, Ruston, Louisiana 71272
  • MR Author ID: 993361
  • ORCID: 0000-0001-6476-0605
  • Email: jdoyle@latech.edu
  • Received by editor(s): April 7, 2016
  • Received by editor(s) in revised form: July 25, 2016
  • Published electronically: November 16, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3265-3288
  • MSC (2010): Primary 37P05; Secondary 37F10, 14H05
  • DOI: https://doi.org/10.1090/tran/7033
  • MathSciNet review: 3766849