Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle

In this paper, we give the upper bound of the number of zeros of Abelian integral I(h)=∮Γhg(x,y)dy−f(x,y)dxI(h)=∮Γhg(x,y)dy−f(x,y)dx, where ΓhΓh is the closed orbit defined by H(x,y)=−x2+x4+y4+rx2y2=hH(x,y)=−x2+x4+y4+rx2y2=h, r≥0r≥0, r≠2r≠2, h∈Σh∈Σ, ΣΣ is the maximal open interval on which the ovals {Γh}{Γh} exist, f(x,y)f(x,y) and g(x,y)g(x,y) are real polynomials in xx and yy of degree at most nn.