## Published In

Transactions of the American Mathematical Society

## Document Type

Pre-Print

## Publication Date

2022

## Subjects

Polynomials -- Dynamical systems

## Abstract

Fix an odd prime p. If r is a positive integer and f is a polynomial with coefficients in Fpr, let Pp,r(f) be the proportion of P1(Fpr) that is periodic with respect to f. We show that as r increases, the expected value of Pp,r(f), as f ranges over quadratic polynomials, is less than 22/(loglogpr). 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.Fix an odd prime p. If r is a positive integer and f is a polynomial with coefficients in Fpr, let Pp,r(f) be the proportion of P1(Fpr) that is periodic with respect to f. We show that as r increases, the expected value of Pp,r(f), as f ranges over quadratic polynomials, is less than 22/(loglogpr). 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.

## Locate the Document

## DOI

10.1090/tran/8634

## Persistent Identifier

https://archives.pdx.edu/ds/psu/37795

## Citation Details

Published as: Garton, D. (2022). Periodic points of polynomials over finite fields. Transactions of the American Mathematical Society.

## Description

This is the author’s version of a work. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document.