On the Action of the Complex Monge–Ampère Operator on Piecewise Linear Functions

The action of the mixed complex Monge–Ampère operator $(h_1,\cdots,h_k)\mapsto dd^ch_1\wedge\cdots\wedge dd^ch_k$ on piecewise linear functions $h_i$ is considered. The language of Monge–Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge–Ampère operator depends only on the product of the functions $h_i$.