Constructive solution of Infinitesimal Hilbert 16th problem

Abelian integrals are integrals of polynomial one-form over a real planar algebraic oval, considered as a (multivalued) function of coefficients of the  form and of the polynomial defining the oval. Their zeros are responsible for appearance of limit cycles in polynomial perturbations of Hamiltonian planar vector fields, and the problem of finding an upper bound for the number of these zeros is called Infinitesimal Hilbert 16th problem. We provide such an upper bound, which is constructive and double exponential in degrees involved.