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

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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)dx, where Γh is the closed orbit defined by H(x,y)=−x2+x4+y4+rx2y2=h, r≥0, r≠2, h∈Σ, Σ is the maximal open interval on which the ovals {Γh} exist, f(x,y) and g(x,y) are real polynomials in x and y of degree at most n.