Year: 2026
Type: preprint
Abstract: We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\mathbb R}^\text{RE}_{\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion--Rolin. We also develop a parallel complex theory for ${\mathbb R}_\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\text{poly}(\log H)$. This generalizes a theorem of Cluckers--Pila--Wilkie who proved the same statement for ${\mathbb R}_\text{an}^\text{pow}$.
Source: arXiv (Cornell University)
Authors: Binyamini, Gal, Carmon, Oded, Novikov, Dmitry
Institution: Weizmann Institute of Science
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213598826