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2019 A finiteness theorem for specializations of dynatomic polynomials
David Krumm
Algebra Number Theory 13(4): 963-993 (2019). DOI: 10.2140/ant.2019.13.963

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 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.

Citation

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David Krumm. "A finiteness theorem for specializations of dynatomic polynomials." Algebra Number Theory 13 (4) 963 - 993, 2019. https://doi.org/10.2140/ant.2019.13.963

Information

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

zbMATH: 07059761
MathSciNet: MR3951585
Digital Object Identifier: 10.2140/ant.2019.13.963

Subjects:
Primary: 37P05
Secondary: 11S15, 37P35

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.13 • No. 4 • 2019
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