The Chebyshev property of some complete hyper-elliptic integrals of the first kind

In this paper, we provide a criterion to determine the Chebyshev property for some complete hyper-elliptic integrals of the first kind. As an application, we study some classes of complete hyper-elliptic integrals of the first kind, which have been considered in the work of Gavrilov and Iliev (Trans Am Math Soc 356:1185–1207). Gavrilov and Iliev showed that the integrals satisfy the Chebyshev property when they belong to exceptional families. Here it is shown that if the Hamiltonian system has only real critical points, for the ovals around the center (1, 0), the corresponding integrals satisfy the Chebyshev property no matter what they belong to exceptional families.