Exponential sums equations and tropical geometry

Abstract

Zilber’s Exponential-Algebraic Closedness Conjecture states that algebraic varieties in $${{\mathbb {C}}}^n \times ({{\mathbb {C}}}^\times )^n$$ C n × ( C × ) n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of $${\textrm{exp}}$$ exp . We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry.