Preparation and Point Counting in Sharply O-minimal Structures

I will describe recent work in progress on logarithmic--exponential preparation theorems in analytically generated sharply o-minimal structures. Our results imply the sharp o-minimality of ℝexp

as well as a uniform version of Wilkie’s conjecture on the density of rational points in definable sets. Our approach is based on the preparation theorem of Lion--Rolin and its extension to the o-minimal setting by van den Dries--Speissegger, and on the theory of complex cells introduced by Binyamini--Novikov.

This is joint work with Gal Binyamini and Dmitry Novikov.