Wilkie’s conjecture for Pfaffian structures

Abstract

We prove an effective form of Wilkie’s conjecture in the structure  generated by restricted sub-Pfaffian functions: the number of  rational points of height $H$ lying in the transcendental part of  such a set grows no faster than some power of $\log H$. Our bounds  depend only on the Pfaffian complexity of the sets involved. As a  corollary we deduce Wilkie’s original conjecture for $\mathbb{R}_{\rm exp}$ in  full generality.