Estimate For The Number Of Zeros Of Abelian Integrals For A Kind Of Quartic Hamiltonians

摘要

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) = y~2 + x~4 - x~2 = 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 E obeys the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the space of functions).