1 results
The Pila–Wilkie theorem for subanalytic families: a complex analytic approach
- Part of
-
- Journal:
- Compositio Mathematica / Volume 153 / Issue 10 / October 2017
- Published online by Cambridge University Press:
- 27 July 2017, pp. 2171-2194
- Print publication:
- October 2017
-
- Article
-
- You have access
- HTML
- Export citation
-
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 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 
$C^{r}$ 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 
$\mathbb{C}^{m}$ 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.
