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

  • Gal Binyamini (a1) and Dmitry Novikov (a2)

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.

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[BN17] Binyamini, G. and Novikov, D., Wilkie’s conjecture for restricted elementary functions , Ann. of Math. (2) 186 (2017), 237275.
[BP89] Bombieri, E. and Pila, J., The number of integral points on arcs and ovals , Duke Math. J. 59 (1989), 337357.
[But12] Butler, L. A., Some cases of Wilkie’s conjecture , Bull. Lond. Math. Soc. 44 (2012), 642660.
[CCL15] 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.
[CPW16] Cluckers, R., Pila, J. and Wilkie, A., Uniform parameterization of subanalytic sets and diophantine applications, Preprint (2016), arXiv:1605.05916.
[DvdD88] Denef, J. and van den Dries, L., p-adic and real subanalytic sets , Ann. of Math. (2) 128 (1988), 79138.
[GKZ94] Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V., Discriminants, resultants, and multidimensional determinants , inMathematics: theory & applications (Birkhäuser Boston, Inc., Boston, MA, 1994).
[Gro87] Gromov, M., Entropy, homology and semialgebraic geometry , Astérisque 145–146 (1987), 225240; Séminaire Bourbaki, Vol. 1985/86.
[GR09] Gunning, R. C. and Rossi, H., Analytic functions of several complex variables (AMS Chelsea Publishing, Providence, RI, 2009), Reprint of the 1965 original.
[Hir64a] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I , Ann. of Math. (2) 79 (1964), 109203.
[Hir64b] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. II , Ann. of Math. (2) 79 (1964), 205326.
[JT12] Jones, G. O. and Thomas, M. E. M., The density of algebraic points on certain Pfaffian surfaces , Q. J. Math. 63 (2012), 637651.
[Pil91] Pila, J., Geometric postulation of a smooth function and the number of rational points , Duke Math. J. 63 (1991), 449463.
[Pil04] Pila, J., Integer points on the dilation of a subanalytic surface , Q. J. Math. 55 (2004), 207223.
[Pil07] Pila, J., The density of rational points on a Pfaff curve , Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), 635645.
[Pil10] Pila, J., Counting rational points on a certain exponential-algebraic surface , Ann. Inst. Fourier (Grenoble) 60 (2010), 489514.
[PW06] Pila, J. and Wilkie, A. J., The rational points of a definable set , Duke Math. J. 133 (2006), 591616.
[Yom87a] Yomdin, Y., Volume growth and entropy , Israel J. Math. 57 (1987), 285300.
[Yom87b] Yomdin, Y., C k -resolution of semialgebraic mappings. Addendum to: ‘Volume growth and entropy’ , Israel J. Math. 57 (1987), 301317.
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The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

  • Gal Binyamini (a1) and Dmitry Novikov (a2)

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