Action of the complex Monge–Ampère operator on piecewise-linear functions and exponential tropical varieties

We consider exponential tropical varieties, which appear as analogues of algebraic tropical varieties when we pass from algebraic varieties to varieties given by zero sets of systems of exponential sums. We describe a construction of exponential tropical varieties arising from the action of the complex Monge–Ampère operator on piecewise-linear functions and show that every such variety can be obtained in this way. As an application, we deduce a criterion for the vanishing of the value of the mixed Monge–Ampère operator. This is an analogue and generalization of the criterion for the vanishing of the mixed volume of convex bodies.