On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x2+y2+ax4+bx2y2+cy4=h, h∈Σ, ac(4ac−b2)≠0, Σ=(0,h1) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n⩾2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.