Resumen de On the Chebyshev Property of a Class of Hyperelliptic Abelian Integrals

Yangjian Sun, Shaoqing Wang, Jiazhong Yang

• This paper aims to demonstrate the Chebyshev property of the linear space V = { 2 i=0 αi h x2i ydx : α0, α1, α2 ∈ R, h ∈ } (which is equivalent to that every function of V has at most 2 zeros, counted with multiplicity), with three hyperelliptic Abelian integrals h x2i ydx (i = 0, 1, 2) as generators, where h is an oval determined by H(x, y) = y2 2 + (x) = h, and (x) is an even polynomial of indefinite degree with real non-Morse critical points. As an application, we can obtain the exact upper bound for the number of zeros of a class of hyperelliptic Abelian integrals related to some planar polynomial Hamiltonian systems with two cusps and a nilpotent center.