The monotonicity of the ratio of hyperelliptic integrals
The weakened Hilbert's 16th problem is to find the isolated zeros of the abelian integral \[ I(h)=\int_{\gamma(h)}^{}f(x,y)dx+g(x,y)dy, \] where \(f(x,y)\) and \(g(x,y)\) are polynomials. The curve \(\gamma(h)\) is a compact component of the level set \(H(x,y)=h\). The aim of this paper is study the monotonicity of the ratio \(P(h)=\frac{I_{1}(h)}{I_{0}(h)}\), where \[ I_{0}(h)=\oint_{\gamma(h)}^{} y dx, \,\,I_{1}(h)=\oint_{\gamma(h)}^{} xy dx. \] The systems under consideration are written in the form \[ \begin{aligned}\dot{x}&=-2y,\dot{y}=x(x-\alpha)(x-\beta)(x-\lambda)(x-1) \\ \dot{x}&=2y, \dot{y}=x(x-\alpha)(x-\beta)(x-\lambda)(x-1).\end{aligned} \] It is shown that \(P(h)\) is monotone for several cases of \(\alpha, \beta,\) and \( \lambda\). Some interesting conjectures are presented.
scientific article; zbMATH DE number 6351416