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 C-r-smooth parametrizations by a variant ofWeierstrass 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 C-r 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 C-m are contained in O(1) complex-algebraic hypersurfaces of degree (logH)(k/(m-k)). This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.