We provide an effective uniform upper bond for the number of zeros of the
first non-vanishing Melnikov function of a polynomial perturbations of a planar
polynomial Hamiltonian vector field. The bound depends on degrees of the field
and of the perturbation, and on the order k of the Melnikov function. The
generic case k=1 was considered by Binyamini, Novikov and Yakovenko
(\cite{BNY-Inf16}). The bound follows from an effective construction of the
Gauss-Manin connection for iterated integrals
