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 ϕ(x) = x2 + t (t)[x], and for every positive integer n let Φn(t,x) denote the n-th dynatomic polynomial of ϕ. Let Gn be the Galois group of Φn over the function field (t), and for c let Gn,c be the Galois group of the specialized polynomial Φn(c,x). It follows from Hilbert’s irreducibility theorem that for fixed n we have GnGn,c for every c outside a thin set En . By earlier work of Morton (for n = 3) and the present author (for n = 4), it is known that En is infinite if n 4. In contrast, we show here that En is finite if n {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 , more than 81% of prime numbers p have the property that the polynomial x2 + c does not have a point of period n in the p-adic field p.

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