## Duke Mathematical Journal

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

#### Abstract

In the context that arose from an old problem of Lang regarding the torsion points on subvarieties of ${\Bbb G}_m^d$, we describe the points that lie in a given variety, are defined over the cyclotomic closure $k^c$ of a number field $k$, and map to a torsion point under a finite projection to ${\Bbb G}_m^d$. We apply this result to obtain a sharp and explicit version of Hilbert's irreducibility theorem over $k^c$. 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 $k^c$. 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

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

Dates
First available in Project Euclid: 24 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1187916269

Digital Object Identifier
doi:10.1215/S0012-7094-07-13934-6

Mathematical Reviews number (MathSciNet)
MR2350852

Zentralblatt MATH identifier
1127.11040

#### Citation

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