  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.
