Gal Binyamini

On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert sixteenth problem

We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small non-conservative 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.

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

We present a complex analytic proof of the Pila-Wilkie theorem for subanalytic sets. In particular, we replace the use of

An explicit linear estimate for the number of zeros of Abelian integrals

C^r

Moment Vanishing of Piecewise Solutions of Linear ODEs

-smooth parametrizations by a variant of Weierstrass division.

Multiplicities of Noetherian Deformations

An Abelian integral is the integral over the level curves of a Hamiltonian H of an algebraic form ?. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the degrees H and ?. Petrov and Khovanskii have shown that this number grows at most linearly with the degree of ?, but gave a purely existential bound. Binyamini, Novikov and Yakovenko have given an explicit bound growing doubly exponentially with the degree.We combine the techniques used in the proofs of these two results, to obtain an explicit bound on the number of zeros of Abelian integrals growing linearly with deg ?.An Abelian integral is the integral over the level curves of a Hamiltonian

Bezout-type theorems for differential fields

H

Wilkie’s conjecture for restricted elementary functions

of an algebraic form

POLYNOMIAL BOUNDS FOR THE OSCILLATION OF SOLUTIONS OF FUCHSIAN SYSTEMS

\omega

FINITENESS PROPERTIES OF FORMAL LIE GROUP ACTIONS

. The infinitesimal Hilbert sixteenth problem calls for the study of the number of zeros of Abelian integrals in terms of the degrees

Intersection multiplicities of Noetherian functions

H