The number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order

A necessary and sufficient condition is given for quasi-homogeneous polynomial Hamiltonian systems having a center. Then it shown that there exists bound on the number of limit cycles bifurcating from period annulus at any order Melnikov functions; explicit expression this in terms (n,k,s1,s2), where n degree perturbation polynomials, k first nonzero higher function, (s1,s2) weight exponent with This extends some known results solves Arnol'd-Hilbert's 16th problem perturbations homogeneous or systems.