On the algebraic structure and the number of zeros of abelian integral for a class of Hamiltonians with degenerate singularities (Q1757112)

The sixteen generators of the abelian integral \[ I(h)=\displaystyle\oint_{\Gamma_{h}}g(x,y)\,dx-f(x,y)\,dy, \] which satisfy eight different Picard-Fuchs equations, respectively, are obtained, where \(\Gamma_{h}\) is a family of closed orbits defined by \[ H(x,y)=ax^4+by^4+cx^8=h,\;h\in\Sigma, \] where \(\Sigma\) 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 for the number of zeros of \(I(h)\) is obtained for the special case \[ f(x,y)=\sum_{0\leq i \leq4k+1=n}a_{i}x^{4k+1-i}y^{i},\quad g(x,y)=\sum_{0\leq i \leq4k+1=n}b_{i}x^{4k+1-i}y^{i}. \]