Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians
When studying the second part of the 16th Hilbert problem one of the interesting problems is that posed by V. I. Arnold: estimate the number of limit cycles for a system on the plane that is a polynomial perturbation of a Hamiltonian polynomial one. Then, due to the Poincaré-Pontryagin method, the problem of estimating the number of cycles (periodic solutions) to the system is reduced to the estimate of the number of zeros of some polynomial Abelian integral along ovals of the unperturbed Hamiltonian for a 1-form depending on the perturbation. In this paper, the authors obtain a linear estimate \(B(n)\leq 7n+5\) in \(n\) (maximal degree of the perturbation) for the Hamiltonian \(H(x,y)=y^{2}/2+U(x)\), \(\deg U(x)=4\), having at least one center singular point.
scientific article; zbMATH DE number 1309592