圆形蛇的利普希茨几何与组合学
Lipschitz geometry and combinatorics of circular snakes
AI总结：
本文研究具圆形链环的实半代数曲面芽（圆形蛇）的利普希茨几何与组合性质，证明Valette链环的规范分解等结果，构造相关组合对象并给出弱外利普希茨分类，揭示二元圆形蛇组合学的独特性。
摘要：
本文探讨了具有圆形链环（同胚于圆$\mathbb{S}^1$）的实半代数曲面芽（或更一般地，在多项式有界o-极小结构中可定义的曲面芽）的利普希茨几何和组合性质。我们定义并研究了所谓圆形蛇的外利普希茨几何，展示了Andrei Gabrielov和Emanoel Souza的论文《异常曲面芽的利普希茨几何与组合学》中对蛇有效的结果在圆形情形下仍然成立。我们证明了圆形蛇的Valette链环可规范分解为有限个线段和节点区，并建立了判定何时可通过移除其一段或Valette链环含于某节点区的Hölder三角形从圆形蛇得到蛇的充要条件。我们构造了与圆形蛇相关的组合对象并证明了该组合对象的实现定理，还给出了圆形蛇的弱外利普希茨分类。最后，我们展示了关于二元圆形蛇组合学的一些结果，这些结果与Gabrielov和Souza工作中所示的对应情形截然不同。
英文摘要：
This paper explores the Lipschitz geometric and combinatorial properties of germs of real semialgebraic surfaces (or, more generally, definable in a polynomially bounded o-minimal structure) with circular link (homeomorphic to the circle $\mathbb{S}^1$). We define and investigate the outer Lipschitz geometry of the so-called circular snakes, showing what results in the paper "Lipschitz geometry and combinatorics of abnormal surface germs" (by Andrei Gabrielov and Emanoel Souza) valid to snakes still holds for the circular case. We prove the existence of a canonical decomposition for the Valette link of a circular snake into finitely many segments and nodal zones and establish some necessary and sufficient criteria to determine when it is possible to obtain a snake from a circular snake by "removing" either one of its segments or a Hölder triangle whose Valette link is contained in one of its nodal zones. We construct a combinatorial object associated with a circular snake and prove a realization theorem for this combinatorial object. We also present a weakly outer Lipschitz classification for circular snakes. Finally, we show some results about the combinatorics of binary circular snakes, which is wildly distinct from the corresponding case shown in the work of Gabrielov and Souza.
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