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 ${\mathbb{R}}_{\exp }$ in full generality.