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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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Periodic points of polynomials over finite fields
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by Derek Garton PDF
Trans. Amer. Math. Soc. 375 (2022), 4849-4871 Request permission


Fix an odd prime $p$. If $r$ is a positive integer and $f$ is a polynomial with coefficients in $\mathbb {F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb {P}^1\left (\mathbb {F}_{p^r}\right )$ that is periodic with respect to $f$. We show that as $r$ increases, the expected value of $P_{p,r}(f)$, as $f$ ranges over quadratic polynomials, is less than $22/\left (\log {\log {p^r}}\right )$. This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.
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Additional Information
  • Derek Garton
  • Affiliation: Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, Oregon
  • MR Author ID: 1024781
  • Email:
  • Received by editor(s): May 18, 2021
  • Received by editor(s) in revised form: December 8, 2021
  • Published electronically: April 21, 2022
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4849-4871
  • MSC (2020): Primary 37P05; Secondary 37P25, 37P35, 11T06, 13B05
  • DOI:
  • MathSciNet review: 4439493