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

Abstract

The sixteen generators of Abelian integral $I(h)=\oint _{\Gamma _h}g(x,y)dx-f(x,y)dy$, which satisfy eight different Picard–Fuchs equations respectively, are obtained, where $\Gamma _h$ is a family of closed orbits defined by $H(x,y)=ax^4+by^4+cx^8=h$, $h\in \Sigma$, $\Sigma$ is the open intervals on which $\Gamma _h$ 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

$$\begin{aligned} f(x,y)=\sum \limits _{0\le i\le 4k+1=n}a_ix^{4k+1-i}y^i,\ \ \ g(x,y)=\sum \limits _{0\le i\le 4k+1=n}b_ix^{4k+1-i}y^i. \end{aligned}$$