Duke Mathematical Journal

Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps)

R. Dvornicich and U. Zannier

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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]

Article information

Duke Math. J., Volume 139, Number 3 (2007), 527-554.

First available in Project Euclid: 24 August 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]
Secondary: 11R18: Cyclotomic extensions 12E25: Hilbertian fields; Hilbert's irreducibility theorem 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]


Dvornicich, R.; Zannier, U. Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps). Duke Math. J. 139 (2007), no. 3, 527--554. doi:10.1215/S0012-7094-07-13934-6. https://projecteuclid.org/euclid.dmj/1187916269

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