A geometric approach to some systems of exponential equations
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.
Keywords
Cite
@article{arxiv.2105.12679,
title = {A geometric approach to some systems of exponential equations},
author = {Vahagn Aslanyan and Jonathan Kirby and Vincenzo Mantova},
journal= {arXiv preprint arXiv:2105.12679},
year = {2024}
}
Comments
34 pages; new remarks 4.10, 5.8, and minor clarifications