Algebra & Number Theory

A finiteness theorem for specializations of dynatomic polynomials

David Krumm

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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 n4. 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.

Article information

Algebra Number Theory, Volume 13, Number 4 (2019), 963-993.

Received: 28 May 2018
Revised: 22 January 2019
Accepted: 22 February 2019
First available in Project Euclid: 18 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37P05: Polynomial and rational maps
Secondary: 11S15: Ramification and extension theory 37P35: Arithmetic properties of periodic points

arithmetic dynamics function fields Galois theory


Krumm, David. A finiteness theorem for specializations of dynatomic polynomials. Algebra Number Theory 13 (2019), no. 4, 963--993. doi:10.2140/ant.2019.13.963.

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