Brunella-Miari向量场中具有两个边的牛顿图的单值奇点
Monodromic singularities of Brunella-Miari vector fields with two edges in the Newton diagram
AI总结：
本文研究了具有两个边的牛顿图的Brunella-Miari向量场的单值奇点，通过去奇异化方法分析了向量场的结构和返回映射的计算。
摘要：
本文聚焦于研究平面解析向量场族中单值奇点的研究，这些向量场的牛顿图恰好由两个边组成。我们首先分析了一个极小模型的去奇异化方案，该模型由仅包含牛顿图顶点对应单项式的向量场X组成。然后我们将该极小模型扩展为所谓的Brunella-Miari向量场X⊂X^[1]，包含所有与牛顿图边上的点相关的单项式。作为第二个扩展，我们考虑包含更高阶项的向量场X^[1]⊂X^[2]，这些更高阶项对应于牛顿图多边形线以上的点。我们方法的关键点是在每次扩展步骤中保持去奇异化几何。我们提供了显式的去奇异化程序，这使得在去奇异化奇点与超曲面多边形相关的情况下能够计算返回映射Π的线性部分。包含若干非平凡例子以说明该方法。
英文摘要：
This work focuses on the study of monodromic singularities in planar analytic families of vector fields whose Newton diagram consists of exactly two edges. We begin by analyzing the desingularization scheme of a minimal model of polynomial vector fields, denoted by ${X}$, which includes only the monomials corresponding to the vertices of the Newton diagram. We then extend this minimal model to the so-called Brunella-Miari vector fields ${X} \subset {X}^{[1]}$, incorporating all monomials associated with points lying on the edges of the Newton diagram. As a second extension, we consider vector fields ${X}^{[1]} \subset {X}^{[2]}$ that include higher-order terms corresponding to points located above the polygonal line in the Newton diagram. The key point of our approach is to preserve the desingularization geometry at each extension step. We provide explicit desingularization procedures, which enable the computation of the linear part of the return map $Π$ in cases where the desingularized singularity is associated with a hyperbolic polycycle. Several nontrivial examples are included to illustrate the method.
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