We consider the structure R RE obtained from (R, <, +, ·) by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height H in the transcendental part of any definable set is bounded by a polynomial in log H. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for R RE -definable sets.

Abstract. We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small nonconservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem.The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection defined over Q (the Gauss-Manin connection) with a quasiunipotent monodromy group.

Let Ω ⊂ R n be a relatively compact domain. A finite collection of real-valued functions on Ω is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila-Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any ε > 0, the number of points of height H in the transcendental part of the set is at most C · H ε where C can be explicitly estimated from the Noetherian parameters and ε.We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety Ag . We thus effectivize the (geometric side of) Pila-Zannier strategy for unlikely intersections in those instances that involve only compact domains.

We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For δ ∈ (0, 1) and a complex cell C we define its holomorphic extension C ⊂ C δ , which is again a complex cell. The hyperbolic geometry of C within C δ provides the class of complex cells with a rich geometric function theory absent in the real case. We use this to prove a complex analog of the cellular decomposition theorem of real tame geometry. In the algebraic case we show that the complexity of such decompositions depends polynomially on the degrees of the equations involved.Using this theory, we refine the Yomdin-Gromov algebraic lemma on C rsmooth parametrizations of semialgebraic sets: we show that the number of C r charts can be taken to be polynomial in the smoothness order r and in the complexity of the set. The algebraic lemma was initially invented in the work of Yomdin and Gromov to produce estimates for the topological entropy of C ∞ maps. For analytic maps our refined version, combined with work of Burguet, Liao and Yang, establishes an optimal refinement of these estimates in the form of tight bounds on the tail entropy and volume growth. This settles a conjecture of Yomdin who proved the same result in dimension two in 1991. A self-contained proof of these estimates using the refined algebraic lemma is given in an appendix by Yomdin.The algebraic lemma has more recently been used in the study of rational points on algebraic and transcendental varieties. We use the theory of complex cells in these two directions. In the algebraic context we refine a result of Heath-Brown on interpolating rational points in algebraic varieties. In the transcendental context we prove an interpolation result for (unrestricted) logarithmic images of subanalytic sets.

Abstract. Following ideas of Arnold and Seigal-Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi. et al.In the second part of the paper we use similar ideas to prove a result on embedding formal diffeomorphisms in one-parameter groups extending a result of Takens. In particular this implies that the results of Arnold and Seigal-Yakovenko are formal consequence of our result for Lie groups.