Study On The Number Of Zeros For Two Classes Hyperelliptic Integrals

In1990, Arnold published a paper in ’Advance in Soviet Mathematics’ ntitled "Ten problems, in:Theory of Singularities and Its Applications", in which the7th problem is divided into two parts. The first part of the7th problem is the weak Hilbert’s16th problem while the second part of the7th problem is about the Chebyshev property of complete hyperelliptic integrals of the first kind. This dissertation is mainly devoted to study the Chebyshev property for three classes of complete hyperelliptic integrals of the first kind and the number of zeros of Abelian integrals under second-order polynomial perturbations of hyperelliptic Hamilton of degree six. These are intimately related to bifurcation theory and qualitative theory of differential equations.Concretely, the main content can be generalized as the following: Firstly, we investigate the the Chebyshev property of complete hyperelliptic integrals of the first kind when the degree of hyperelliptic Hamilton is five. To the best of our knowledge, Gavrilov and Iliev [81] seem to be the first to study the Chebyshev property of complete hyperelliptic integrals of the first kind. They obtained that there exist exceptional families of ovals of the hyperelliptic Hamilton of degree five such that the corresponding complete hyperelliptic integrals of the first kind is Chebyshev. Their proof on Chebyshev property is based on the Rieman bilinear relations on differentials of the first kind together with the fact that a Jacobian variety with its polarization cannot be a direct product of principally polarized Abelian varieties. Motivated by their work, we study the complete hyperelliptic integrals of the first kind for three classes of degenerate families of ovals in [81], which are the boundary points of the bifurcation diagram of hyperelliptic Hamilton system of degree five. It turns out that one class of the continuous family of ovals is exceptional and the others are not exceptional. By using real analytic methods and symbolic computations as well as the asymptotic expansions of Abelian integrals we prove that the three classes of complete hyperelliptic integrals are Chebyshev. This result reveals that exceptional families is not the necessary condition such that the corresponding complete hyperelliptic integrals of the first kind is Chebyshev.Secondly, we study the number of zeros of Abelian integrals for the hyperellip-tic Hamilton system of degree six under quadratic polynomial perturbations, which, equivalently, can be converted to the study of the monotonicity of the ratio of two hyperelliptic Abelian integrals of degree six. We first present a complete topolog-ical classification of level curves for the hyperelliptic Hamilton of degree six that has only real critical points. Then, in the case of the integral domain are compact components surrounding only a unique center, we further study the monotonicity of the ratio of two hyperelliptic integrals, and finally, we obtain some sufficient conditions under which the ratio of the two Abelian integrals is monotone or non-monotone. In other words, we find some sufficient conditions which ensure that the number of zeros of the Abelian integrals for the hyperelliptic Hamilton system of degree six under quadratic polynomial perturbations is exact one or more than one.