关于椭圆奇点链的典范多项式
On a canonical polynomial for links of elliptic singularities
AI总结：
本文研究了正规曲面奇点的典范多项式与椭圆序列这两个拓扑不变量之间的关系，证明了多项式指数决定椭圆序列，并通过图的扩展定义了好扩展以反映其标志结构。
摘要：
典范多项式是与正规曲面奇点相关的多变量拓扑Poincaré级数的重要输出。它可以被视为链的Seiberg--Witten不变量的多变量多项式推广。在椭圆芽的情况下，人们考虑了另一个关键拓扑不变量，即椭圆序列，它反映了椭圆芽的特定结构并指导了它们的若干性质。在本文中，我们研究了这两个对象之间的关系。首先，我们描述了典范多项式指数的结构，并证明它们决定了椭圆序列。对于逆问题，我们通过自然扩展椭圆芽的图来考虑其归纳设定，并比较相应的指数集合。这引出了好扩展的定义，它可以通过相应典范多项式的包含类型公式来刻画。这以兼容的方式反映了椭圆序列的“标志结构”。
英文摘要：
The canonical polynomial is an important output of the multivariable topological Poincaré series associated with a normal surface singularity. It can be considered as a multivariable polynomial generalization of the Seiberg--Witten invariant of the link. In the case of elliptic germs, another key topological invariant was considered, the elliptic sequence, which mirrors the specific structure of the elliptic germs and guides several properties of them. In this note we study the relationship of these two objects. First of all, we describe the structure of the exponents of the canonical polynomial and prove that they determine the elliptic sequence. For the converse problem, we consider an inductive setup of elliptic germs via natural extension of their graphs and compare the corresponding sets of exponents. This leads to the definition of a good extension which can be characterized by an inclusion type formula for the corresponding canonical polynomials. This reflects in a compatible way the `flag structure' of the elliptic sequence.
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