The exact bounds on the number of zeros of complete hyperelliptic integrals of the first kind

In this paper we study three classes of complete hyperelliptic integrals of the first kind, which are some degenerate subfamilies of a family considered in the work of Gavrilov and Iliev. It is shown that the three classes of complete hyperelliptic integrals are Chebyshev, and the exact bounds on the number of zeros of these Abelian integrals are one. This result reveals that there exist degenerate subfamilies of ovals of the hyperelliptic Hamiltonian which are not exceptional families proposed by Gavrilov and Iliev, but the corresponding complete hyperelliptic integrals of the first kind still satisfy the Chebyshev property.