Efficiently Detecting Torsion Points and Subtori

Suppose X is the complex zero set of a finite collection of polynomials in Z[x1, ..., xn]. We show that deciding whether X contains a point all of whose coordinates are dth roots of unity can be done within 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 SiegelLandau zeroes. Furthermore, we give a similar unconditional complexity upper bound for n=1. Finally, letting T be any algebraic subgroup of (C) we show that deciding X ? ⊇ 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.