Exponential tropical varieties and complex Monge-Ampere operator

TL;DR

This paper introduces exponential tropical varieties (ETV) as complex analogues of tropical varieties, constructs their algebraic structure, and connects them with the complex Monge-Ampere operator, providing new criteria and applications in algebraic geometry.

Contribution

It defines the ring of ETV, shows their relation to the Monge-Ampere operator, and develops criteria for zero values of mixed Monge-Ampere operators, extending tropical geometry concepts.

Findings

ETV form a ring containing algebraic tropical varieties as a subring

All ETV can be obtained via the complex Monge-Ampere operator acting on piecewise linear functions

Provides a criterion for the zero value of mixed Monge-Ampere operators

Abstract

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 The proof is the modification of A. Khovanskii's unpublished proof of the corresponding theorem on mixed volumes.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.