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

Part of: Arithmetic algebraic geometry Number theory: Connections with logic Model theory

Published online by Cambridge University Press:  27 July 2017

Gal Binyamini  and
Dmitry Novikov
Gal Binyamini
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel email gal.binyamini@weizmann.ac.il
Dmitry Novikov
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel email dmitry.novikov@weizmann.ac.il
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Abstract

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.

Keywords

Bombieri–Pila theoremrational pointsdefinable setsinterpolation determinants

MSC classification

Primary: 11G99: None of the above, but in this section
Secondary: 14P15: Real analytic and semianalytic sets 03C64: Model theory of ordered structures; o-minimality 11U09: Model theory
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 10 , October 2017 , pp. 2171 - 2194
Copyright
© The Authors 2017 

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