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We prove that the number of limit cycles generated by a smallnon-conservative perturbation of a Hamiltonian polynomial vector field on theplane, is bounded by a double exponential of the degree of the fields. Thissolves the long-standing tangential Hilbert 16th problem. The proof uses onlythe fact that Abelian integrals of a given degree are horizontal sections of aregular flat meromorphic connection (Gauss-Manin connection) with aquasiunipotent monodromy group.

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem