A finiteness theorem for specializations of dynatomic polynomials

Abstract

Let $t$ and $x$ be indeterminates, let $\varphi \left(x\right)=x^2+t\in \mathbb{Q}\left(t\right)\left[x\right]$, and for every positive integer $n$ let ${\mathrm{\Phi }}_n\left(t,x\right)$ denote the $n$-th dynatomic polynomial of $\varphi$. Let $G_n$ be the Galois group of ${\mathrm{\Phi }}_n$ over the function field $\mathbb{Q}\left(t\right)$, and for $c\in \mathbb{Q}$ let $G_{n,c}$ be the Galois group of the specialized polynomial ${\mathrm{\Phi }}_n\left(c,x\right)$. 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 $E_n\subset \mathbb{Q}$. 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 $c\in \mathbb{Q}$, 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 ${\mathbb{Q}}_p$.