  Sometimes it is possible to extend some using Newton polyhedra computations
in algebraic geometry from polynomials to exponential sums. For this purpose it
is useful to consider analogues of tropical varieties in complex space. These
analogues are called exponential tropical varieties (ETV). We construct the
ring of ETV. Algebraic tropical varieties form the subring of the ring of ETV.
In this paper we connect ETV with the complex Monge-Ampere operator action on
the space of piecewise linear functions in complex vector space. We show that
all ETV arise as results of such operator action. We give some applications of
this connection. One of the applications is a criterion for zero value of a
mixed Monge-Ampere operator. This criterion is the modification of the
criterion for zero value of a mixed volume of convex bodies. The proof is the
modification of A. Khovanskii's unpublished proof of the corresponding theorem
on mixed volumes. In the part 1 we give the definition of ETV and detail
statements of theorems (without proofs). In the part 2 we prove the theorems on
the action of the Monge-Ampere operator.

Algebraic Geometry