The number of zeros of the sum of Abelian integrals satisfying two-dimensional Picard-Fuchs equations

This paper is primarily devoted to developing a new criterion to show the upper bound for the number of zeros of the sum of Abelian integrals satisfying two-dimensional Picard-Fuchs equations, which is a generalization of the method by Gavrilov and Iliev (2003) [13]. The second goal of this paper is to investigate, using our new criterion, the hyperelliptic Abelian integrals related to the period annuli of a Hamiltonian system $\dot{x}=-y,\dot{y}=T_{n+1}^{\prime }\left(x\right)$ under the polynomial deformation of degree n, where $T_{n+1}\left(x\right)=\cos \left(\right(n+1)\arccos x)$ is the Chebyshev polynomial of the first kind, and to show that for every period annulus of such system, the number of zeros of the hyperelliptic Abelian integral is no more than $O\left(n^2\right)$, which is novel up to now.