Abstract In this paper, we develop a new method to estimate the smallest upper bound on the number of isolated zeros of Abelian integrals, and give an algebraic criterion for the case of Abelian integrals along energy level ovals of potential systems. As applications of our main result, we first obtain a criterion guaranteeing that two Abelian integrals have the Chebyshev property with accuracy one and give an example of hyperelliptic Abelian integrals. Then we get a criterion that any nontrivial linear combination of three Abelian integrals has at most two isolated zeros. Using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non-algebraic Abelian integral along the oval ( y 2 + 1 2 ) 1 x 2 + ln ⁡ | x | − 1 2 = h with 0 h + ∞ , which corresponds to the perturbation of a quadratic reversible system with two centers. Moreover, we derive all possible configurations of limit cycles from the Poincare bifurcation of this perturbation. To our knowledge, these problems can not be solved by other known approaches.

中文翻译：

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阿贝尔积分的零点个数的最小上界

摘要 在本文中，我们开发了一种估计阿贝尔积分孤立零点数的最小上界的新方法，并给出了阿贝尔积分沿势系统能级椭圆的情况的代数判据。作为我们主要结果的应用，我们首先获得一个准则，保证两个阿贝尔积分具有精度为 1 的切比雪夫性质，并给出一个超椭圆阿贝尔积分的例子。然后我们得到一个标准，即三个阿贝尔积分的任何非平凡线性组合至多有两个孤立的零点。使用这个标准，我们证明了沿椭圆 (y 2 + 1 2 ) 1 x 2 + ln ⁡ | 的非代数阿贝尔积分的孤立零点数的最小上限是 2。× | − 1 2 = h 0 h + ∞ , 这对应于具有两个中心的二次可逆系统的扰动。此外，我们从这种扰动的庞加莱分岔中推导出所有可能的极限环配置。据我们所知，这些问题无法通过其他已知方法解决。