Exponential Sums Equations and Tropical Geometry

Abstract

We show a case of Zilber's Exponential-Algebraic Closedness Conjecture, establishing that the conjecture holds for varieties which split as the product of a linear subspace of the additive group $\mathbb{C}^n$ and an algebraic subvariety of the multiplicative group $(\mathbb{C}^\times)^n$. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required stronger assumptions on the variety such as the linear space being defined over the real numbers. The proofs use the theory of amoebas and tropical geometry.