Estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians with two centers

In this paper, we give the upper bound of the number of zeros of Abelian integral I(h)=∮ΓhP(x,y)dx-Q(x,y)dyI(h)=∮ΓhP(x,y)dx-Q(x,y)dy, where ΓhΓh is the closed orbit defined by H(x,y)=-x2+λx4+y4=hH(x,y)=-x2+λx4+y4=h, λ>0λ>0, h∈Σh∈Σ; ΣΣ is the maximal open interval on which the ovals {Γh}{Γh} exist; P(x,y)P(x,y) and Q(x,y)Q(x,y) are real polynomials in xx and yy of degree at most nn.