单值性与切心问题
Monodromy and Tangential Center Problems
摘要：
我们考虑由平面哈密顿系统的扰动引起的阿贝尔积分族。切向中心焦点问题要求这些积分为零的条件。这个问题是密切相关的monodromy问题，它问当monodromy的消失循环产生的整体同源的水平曲线的哈密尔顿。我们解决这两个问题的情况下，当哈密顿是超椭圆的。作为副积，我们解决了Gavrilov和Movasati定义的“0维阿贝尔积分“的相应问题。
英文摘要：
We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center focus problem asks for the conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both these questions for the case when the Hamiltonian is hyperelliptic. As a side-product, we solve the corresponding problems for the "0-dimensional Abelian integrals" defined by Gavrilov and Movasati.
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