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

In this paper, we give a lower upper bound of the number of zeros of part of the Abelian integral I(h) = ∫δ_(h) P(x, y)dx + Q(x, y)dy, h ∈ ∑, where δ(h) is an oval contained in the level set {H(x, y) = y² + x⁴ - x² = h}, P(x, y), Q(x, y) are real polynomials of x and y with degree not greater than n, ∑ is the maximal interval of the existence of the ovals {δ(h)}. The corresponding vector space of the Abelian integral I(h) defined on the open interval ∑ obeys the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the space of functions).