The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of Cʳ -smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a Cʳ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height H in a compact piece of a complex-analytic set of dimension k in Cᵐ are contained in $O(1)$ complex-algebraic hypersurfaces of degree (log H)k/(m-k) . This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.