A necessary and sufficient condition is given for quasi-homogeneous polynomial Hamiltonian systems having a center. Then it is shown that there exists a bound on the number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order of Melnikov functions; and the explicit expression of this bound is given in terms of $(n,k,s_1,s_2)$, where n is the degree of perturbation polynomials, k is the order of the first nonzero higher order Melnikov function, and $(s_1,s_2)$ is the weight exponent of quasi-homogeneous Hamiltonian with center. This extends some known results and solves the Arnol'd-Hilbert's 16th problem for the perturbations of homogeneous or quasi-homogeneous polynomial Hamiltonian systems.

给出了具有中心的准齐次多项式哈密顿系统的充要条件。然后表明，在任何Melnikov函数阶上，从准齐次哈密顿系统的周期圆环分叉的极限环数存在一个界。该界限的显式表示为$（ñ，ķ，s_{\mathrm{1个}}，s_2）$，其中n是摄动多项式的阶数，k是第一个非零高阶Melnikov函数的阶，以及$（s_{\mathrm{1个}}，s_2）$是具有中心的准齐次哈密顿量的重量指数。这扩展了一些已知的结果，并解决了Arnol'd-Hilbert的第16个问题，涉及齐次或准齐次多项式哈密顿系统的摄动。