On the Algebraic Structure and the Number of Zeros of Abelian Integral for a Class of Hamiltonians with Degenerate Singularities.

The sixteen generators of Abelian integral I(h)=∮Γhg(x,y)dx-f(x,y)dy, which satisfy eight different Picard-Fuchs equations respectively, are obtained, where Γh is a family of closed orbits defined by H(x,y)=ax4+by4+cx8=h, h∈Σ, Σ is the open intervals on which Γh is defined, and f(x, y) and g(x, y) are real polynomials in x and y of degree n. Moreover, an upper bound of the number of zeros of I(h) is obtained for a special case f(x,y)=∑0≤i≤4k+1=naix4k+1-iyi,g(x,y)=∑0≤i≤4k+1=nbix4k+1-iyi. [ABSTRACT FROM AUTHOR]