A finiteness theorem for specializations of dynatomic polynomials

Krumm, David

doi:10.2140/ant.2019.13.963

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A finiteness theorem for specializations of dynatomic polynomials

David Krumm

Vol. 13 (2019), No. 4, 963–993

DOI: 10.2140/ant.2019.13.963

Abstract

Let $t$ and $x$ be indeterminates, let $ϕ (x) = x^{2} + t \in ℚ (t) [x]$ , and for every positive integer $n$ let $Φ_{n} (t, x)$ denote the $n$ -th dynatomic polynomial of $ϕ$ . Let $G_{n}$ be the Galois group of $Φ_{n}$ over the function field $ℚ (t)$ , and for $c \in ℚ$ let $G_{n, c}$ be the Galois group of the specialized polynomial $Φ_{n} (c, x)$ . It follows from Hilbert’s irreducibility theorem that for fixed $n$ we have $G_{n} ≅ G_{n, c}$ for every $c$ outside a thin set $E_{n} \subset ℚ$ . By earlier work of Morton (for $n = 3$ ) and the present author (for $n = 4$ ), it is known that $E_{n}$ is infinite if $n \leq 4$ . In contrast, we show here that $E_{n}$ is finite if $n \in {5, 6, 7, 9}$ . As an application of this result we show that, for these values of $n$ , the following holds with at most finitely many exceptions: for every $c \in ℚ$ , more than $81 %$ of prime numbers $p$ have the property that the polynomial $x^{2} + c$ does not have a point of period $n$ in the $p$ -adic field $ℚ_{p}$ .

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Keywords

arithmetic dynamics, function fields, Galois theory

Mathematical Subject Classification 2010

Primary: 37P05

Secondary: 11S15, 37P35

Milestones

Received: 28 May 2018

Revised: 22 January 2019

Accepted: 22 February 2019

Published: 6 May 2019

Authors

David Krumm
Reed College Portland, OR United States

Vol. 13, No. 4, 2019