This work concerns with polynomial families of real planar vector fields having a monodromic nilpotent singularity. The families considered are those for which the centers are characterized by the existence of a formal inverse integrating factor vanishing at the singularity with a leading term of minimum (1,n)-quasihomogeneous weighted degree, being n the Andreev number of the singularity. These families strictly include the case n = 2 and also the ℤ2-equivariant families. In some cases for such families we solve, under additional assumptions, the local Hilbert 16th problem giving global bounds on the maximum number of limit cycles that can bifurcate from the singularity under perturbations within the family. Several examples are given.

中文翻译：

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一些幂零奇点的中心循环性，包括ℤ2-等变类

这项工作涉及具有单调幂零奇点的实平面矢量场的多项式族。所考虑的族是那些中心的特征在于存在形式逆积分因子的族，该因子在奇点处消失，前导项为最小值(1,n)- 准齐次加权度，被n奇点的 Andreev 数。这些家庭严格包括案件n = 2还有ℤ2- 等变的家庭。在某些情况下，对于此类族，我们在附加假设下求解局部 Hilbert 16th 问题，该问题给出了在族内扰动下可以从奇点分叉的最大极限环数的全局界限。给出了几个例子。