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

The finite generators of Abelian integral I ( h ) = ∮ Γ h f ( x , y ) d x − g ( x , y ) d y are obtained, where Γ h is a family of closed ovals defined by H ( x , y ) = x 2 + y 2 + a x 4 + b x 2 y 2 + c y 4 = h , h ∈ Σ , a c ( 4 a c − b 2 ) ≠ 0 , Σ = ( 0 , h 1 ) is the open interval on which Γ h is defined, f ( x , y ) , g ( x , y ) are real polynomials in x and y with degree 2 n + 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 .