Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points

In this paper, we study the number of isolated zeros of Abelian integrals associated to system x˙=y,y˙=−x3(x2−1)2 under the perturbations of ϵ(α0+α1x2+α2x4+α3x6)y∂∂y, where 0 < |ϵ| ≪ 1 and αi ∈ R  (i=0,1,2,3i=0,1,2,3). We find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used. The bounds and sharp bounds are obtained.