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We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of
-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
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
in a compact piece of a complex-analytic set of dimension
are contained in
complex-algebraic hypersurfaces of degree
. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.
We prove analogs of the Bezout and the Bernstein–Kushnirenko–Khovanskii theorems for systems of algebraic differential conditions over differentially closed fields. Namely, given a system of algebraic conditions on the first
derivatives of an
-tuple of functions, which admits finitely many solutions, we show that the number of solutions is bounded by an appropriate constant (depending singly-exponentially on
) times the volume of the Newton polytope of the set of conditions. This improves a doubly-exponential estimate due to Hrushovski and Pillay. We illustrate the application of our estimates in two diophantine contexts: to counting transcendental lattice points on algebraic subvarieties of semi-abelian varieties, following Hrushovski and Pillay; and to counting the number of intersections between isogeny classes of elliptic curves and algebraic varieties, following Freitag and Scanlon. In both cases we obtain bounds which are singly-exponential (improving the known doubly-exponential bounds) and which exhibit the natural asymptotic growth with respect to the degrees of the equations involved.
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