A finiteness theorem for specializations of dynatomic polynomials

TL;DR

This paper proves that for certain degrees, the set of specializations where dynatomic polynomials do not have the generic Galois group is finite, leading to density results about periodic points of quadratic polynomials over p-adic fields.

Contribution

It establishes finiteness of the exceptional set of specializations for dynatomic polynomials when n is 5, 6, 7, or 9, extending previous infinite cases.

Findings

Finiteness of the exceptional set for n=5,6,7,9

Most primes lack points of period n for quadratic polynomials over Q_p

Over 81% of primes have no period n points for x^2+c

Abstract

Let $t$ and $x$ be indeterminates, let $\phi(x)=x^2+t\in\mathbb Q(t)[x]$, and for every positive integer $n$ let $\Phi_n(t,x)$ denote the $n^{\text{th}}$ dynatomic polynomial of $\phi$. Let $G_n$ be the Galois group of $\Phi_n$ over the function field $\mathbb Q(t)$, and for $c\in\mathbb Q$ let $G_{n,c}$ be the Galois group of the specialized polynomial $\Phi_n(c,x)$. 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 $n\in\{5,6,7,9\}$. 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 $c\in\mathbb Q$, 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 $\mathbb Q_p$.