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15 September 2007 Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps)
R. Dvornicich, U. Zannier
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Duke Math. J. 139(3): 527-554 (15 September 2007). DOI: 10.1215/S0012-7094-07-13934-6

Abstract

In the context that arose from an old problem of Lang regarding the torsion points on subvarieties of Gmd, we describe the points that lie in a given variety, are defined over the cyclotomic closure kc of a number field k, and map to a torsion point under a finite projection to Gmd. We apply this result to obtain a sharp and explicit version of Hilbert's irreducibility theorem over kc. Concerning the arithmetic of dynamics in one variable, we obtain by related methods a complete description of the polynomials having an infinite invariant set contained in kc. In particular, we answer a number of long-standing open problems posed by W. Narkiewicz and which he eventually collected explicitly in the book [N2]

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R. Dvornicich. U. Zannier. "Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps)." Duke Math. J. 139 (3) 527 - 554, 15 September 2007. https://doi.org/10.1215/S0012-7094-07-13934-6

Information

Published: 15 September 2007
First available in Project Euclid: 24 August 2007

zbMATH: 1127.11040
MathSciNet: MR2350852
Digital Object Identifier: 10.1215/S0012-7094-07-13934-6

Subjects:
Primary: 11G10
Secondary: 11R18, 12E25, 37F10

Rights: Copyright © 2007 Duke University Press

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Vol.139 • No. 3 • 15 September 2007
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