零维无穷小第16希尔伯特问题
The infinitesimal 16th Hilbert problem in dimension zero
摘要：
我们研究无穷小的第16希尔伯特问题的模拟零维。给出了相应的阿贝尔积分（代数函数）零点个数的上界，研究了定义在$\mathbb{Q}$上的Abel积分$I(t)$的消失与其算术性质之间的关系.最后，我们给出了阿贝尔积分恒为零的充要条件。
英文摘要：
We study the analogue of the infinitesimal 16th Hilbert problem in dimension zero. Lower and upper bounds for the number of the zeros of the corresponding Abelian integrals (which are algebraic functions) are found. We study the relation between the vanishing of an Abelian integral $I(t)$ defined over $\mathbb{Q}$ and its arithmetic properties. Finally, we give necessary and sufficient conditions for an Abelian integral to be identically zero.
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