#### Vol. 13, No. 4, 2019

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A finiteness theorem for specializations of dynatomic polynomials

### David Krumm

Vol. 13 (2019), No. 4, 963–993
##### Abstract

Let $t$ and $x$ be indeterminates, let $\varphi \left(x\right)={x}^{2}+t\in ℚ\left(t\right)\left[x\right]$, and for every positive integer $n$ let ${\Phi }_{n}\left(t,x\right)$ denote the $n$-th dynatomic polynomial of $\varphi$. Let ${G}_{n}$ be the Galois group of ${\Phi }_{n}$ over the function field $ℚ\left(t\right)$, and for $c\in ℚ$ let ${G}_{n,c}$ be the Galois group of the specialized polynomial ${\Phi }_{n}\left(c,x\right)$. It follows from Hilbert’s irreducibility theorem that for fixed $n$ we have ${G}_{n}\cong {G}_{n,c}$ 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 $n\le 4$. In contrast, we show here that ${E}_{n}$ is finite if $n\in \left\{5,6,7,9\right\}$. 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