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

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

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 [email protected]
Dmitry Novikov
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel email [email protected]
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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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References

Binyamini, G. and Novikov, D., Wilkie’s conjecture for restricted elementary functions , Ann. of Math. (2) 186 (2017), 237–275.Google Scholar
Bombieri, E. and Pila, J., The number of integral points on arcs and ovals , Duke Math. J. 59 (1989), 337–357.Google Scholar
Butler, L. A., Some cases of Wilkie’s conjecture , Bull. Lond. Math. Soc. 44 (2012), 642–660.Google Scholar
Cluckers, R., Comte, G. and Loeser, F., Non-Archimedean Yomdin–Gromov parametrizations and points of bounded height , Forum Math. Pi 3 (2015), e5, doi:10.1017/fmp.2015.4.Google Scholar
Cluckers, R., Pila, J. and Wilkie, A., Uniform parameterization of subanalytic sets and diophantine applications, Preprint (2016), arXiv:1605.05916.Google Scholar
Denef, J. and van den Dries, L., p-adic and real subanalytic sets , Ann. of Math. (2) 128 (1988), 79–138.Google Scholar
Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants , inMathematics: theory & applications (Birkhäuser Boston, Inc., Boston, MA, 1994).Google Scholar
Gromov, M., Entropy, homology and semialgebraic geometry , Astérisque 145–146 (1987), 225–240; Séminaire Bourbaki, Vol. 1985/86.Google Scholar
Gunning, R. C. and Rossi, H., Analytic functions of several complex variables (AMS Chelsea Publishing, Providence, RI, 2009), Reprint of the 1965 original.Google Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I , Ann. of Math. (2) 79 (1964), 109–203.Google Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II , Ann. of Math. (2) 79 (1964), 205–326.Google Scholar
Jones, G. O. and Thomas, M. E. M., The density of algebraic points on certain Pfaffian surfaces , Q. J. Math. 63 (2012), 637–651.CrossRefGoogle Scholar
Pila, J., Geometric postulation of a smooth function and the number of rational points , Duke Math. J. 63 (1991), 449–463.CrossRefGoogle Scholar
Pila, J., Integer points on the dilation of a subanalytic surface , Q. J. Math. 55 (2004), 207–223.Google Scholar
Pila, J., The density of rational points on a Pfaff curve , Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), 635–645.CrossRefGoogle Scholar
Pila, J., Counting rational points on a certain exponential-algebraic surface , Ann. Inst. Fourier (Grenoble) 60 (2010), 489–514.Google Scholar
Pila, J. and Wilkie, A. J., The rational points of a definable set , Duke Math. J. 133 (2006), 591–616.Google Scholar
Yomdin, Y., Volume growth and entropy , Israel J. Math. 57 (1987), 285–300.Google Scholar
Yomdin, Y., C k -resolution of semialgebraic mappings. Addendum to: ‘Volume growth and entropy’ , Israel J. Math. 57 (1987), 301–317.Google Scholar