A geometric approach to some systems of exponential equations

TL;DR

This paper proves a special case of Zilber's Exponential Algebraic Closedness conjecture for certain varieties, using complex geometric methods to analyze intersections with exponential maps in semiabelian varieties.

Contribution

It establishes the conjecture for varieties with dominant projection to the exponential domain in abelian varieties and algebraic tori, and describes the structure of zero-dimensional intersections.

Findings

Proved the conjecture for varieties with dominant projection in abelian varieties and tori.

Parametrized large intersection points via the period lattice and analytic maps.

Provided a complex geometric approach contrasting previous real analytic proofs.

Abstract

Zilber's Exponential Algebraic Closedness conjecture (also known as Zilber's Nullstellensatz) gives conditions under which a complex algebraic variety should intersect the graph of the exponential map of a semiabelian variety. We prove the special case of the conjecture where the variety has dominant projection to the domain of the exponential map, for abelian varieties and for algebraic tori. Furthermore, in the situation where the intersection is 0-dimensional, we exhibit structure in the intersection by parametrizing the sufficiently large points as the images of the period lattice under a (multivalued) analytic map. Our approach is complex geometric, in contrast to a real analytic proof given by Brownawell and Masser just for the case of algebraic tori.