A weak version of the Strong Exponential Closure
Paola D'Aquino, Antongiulio Fornasiero, Giuseppina Terzo
TL;DR
The paper investigates Strong Exponential Closure for the complex exponential field under Schanuel's Conjecture. It proves that for irreducible $V \subseteq \mathbb C^n \times (\mathbb C^{*})^n$ with $\dim V = n$ and dominant projections, there exists a Zariski-dense set of generic points $(\bar z, e^{\bar z}) \in V$ with $t.d._{\mathbb Q}(\bar z, e^{\bar z}) = n$, thereby providing broad instances of SEC in the complex setting. The approach combines definable dimension theory, properties of rotund/free varieties, and a BM-type solvability criterion, leveraging Schanuel's Conjecture to rule out non-generic configurations. The results advance the understanding of exponential-algebraic geometry and support quasi-minimality-type phenomena for $(\mathbb C, \exp)$ under SC, with implications for model theory of exponential fields.
Abstract
Assuming Schanuel's Conjecture we prove that for any variety V over the algebraic closure over the rational numbers, of dimension n and with dominant projections, there exists a generic point in V. We obtain in this way many instances of the Strong Exponential Closure introduced by Zilber.