Cyclotomic Diophantine Problems (Hilbert Irreducibility and Invariant Sets for Polynomial Maps)

In the context which arose from an old problem of Lang regarding the torsion points on subvarieties of $\G_m^d$, 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 $\G_m^d$. 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 longstanding open problems by Narkiewicz, which he eventually collected explicitly in the book [N].