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

This paper proves several results. First, there is an irreducibility theorem for the lifting of torsion points on a finite cover of G_m^n. Then this is applied to prove a strong form of Hilbert Irreducibility over cyclotomic fields; for instance if f(x,y) is irreducible, under a necessary simple condition, f(a,y) remains irreducible over a maximal cyclotomic extension for every root of unity a. Third, polynomial maps with infinitely many preperiodic points over a maximal cyclotomic field are completely characterized. In particular, the results answer completely some open questions of Narkiewicz