Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Preperiodic portraits for unicritical polynomials over a rational function field


Author: John R. Doyle
Journal: Trans. Amer. Math. Soc. 370 (2018), 3265-3288
MSC (2010): Primary 37P05; Secondary 37F10, 14H05
DOI: https://doi.org/10.1090/tran/7033
Published electronically: November 16, 2017
MathSciNet review: 3766849
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 37P05, 37F10, 14H05

Retrieve articles in all journals with MSC (2010): 37P05, 37F10, 14H05


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

Keywords: Preperiodic points, abc-theorem, unicritical polynomials
Received by editor(s): April 7, 2016
Received by editor(s) in revised form: July 25, 2016
Published electronically: November 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society