The number of maximal torsion cosets in subvarieties of tori

We present sharp bounds on the number of maximal torsion cosets in a subvariety of the complex algebraic torus $\Gm^n$.Our first main result gives a bound in terms of the degree of the defining polynomials.A second result gives a bound in terms of the toric degree of the subvariety.As a consequence, we prove the conjectures of Ruppert and of Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively