Focal values of plane cubic centers

We prove that the vanishing of 11 focal values is not sufficient to ensure that a plane cubic system has a center.Comment: 5 pages. Programs used can be found at http://www.stud.uni-hannover.de/~kroeker/centerfocus/index.html or at http://sourceforge.net/projects/centerfocus

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 by 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 tangential 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 (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio