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 ${\mathbb{G}}_m^d$, 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 ${\mathbb{G}}_m^d$. 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]