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 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]