  Suppose X is the complex zero set of a finite collection of polynomials in
Z[x_1,...,x_n]. We show that deciding whether X contains a point all of whose
coordinates are d_th roots of unity can be done within NP^NP (relative to the
sparse encoding), under a plausible assumption on primes in arithmetic
progression. In particular, our hypothesis can still hold even under certain
failures of the Generalized Riemann Hypothesis, such as the presence of
Siegel-Landau zeroes. Furthermore, we give a similar (but UNconditional)
complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of
(C^*)^n we show that deciding whether X contains T is coNP-complete (relative
to an even more efficient encoding),unconditionally. We thus obtain new
non-trivial families of multivariate polynomial systems where deciding the
existence of complex roots can be done unconditionally in the polynomial
hierarchy -- a family of complexity classes lying between PSPACE and P,
intimately connected with the P=?NP Problem. We also discuss a connection to
Laurent's solution of Chabauty's Conjecture from arithmetic geometry.
