  Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's
Nullstellensatz) gives conditions under which a complex algebraic variety
should intersect the graph of the exponential map of a semiabelian variety.
  We prove the special case of the conjecture where the variety has dominant
projection to the domain of the exponential map, for abelian varieties and for
algebraic tori. Furthermore, in the situation where the intersection is
0-dimensional, we exhibit structure in the intersection by parametrizing the
sufficiently large points as the images of the period lattice under a
(multivalued) analytic map. Our approach is complex geometric, in contrast to a
real analytic proof given by Brownawell and Masser just for the case of
algebraic tori.
