In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl. 499 (2017)], the authors proved that when parameter [Formula: see text] is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when [Formula: see text], the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles.

中文翻译：

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具有幂零线性部分的 Codimension-3 奇点展开的阿贝尔积分

在本文中，我们研究了具有幂零线性部分的 codimension-3 平面奇点展开的最大极限环数。在 [J. 数学。肛门。应用程序。499（2017）]，作者证明当参数[公式：见正文]是有理的时，相应的问题可以转化为求解半代数系统。同时指出，当【公式：见正文】时，按方法会出现对数函数，导致无法解决问题。在本文中，我们使用一些技术来避免对数函数的出现，并得到相应的系统最多产生两个极限环。