Bi-Lipschitz contact invariance of rank

Multiplicity and degree as bi-Lipschitz invariants for complex sets

We study invariance of multiplicity of complex analytic germs and degree of complex affine sets under outer bi-Lipschitz transformations (outer bi-Lipschitz homeomorphims of germs in the first case and outer bi-Lipschitz homeomorphims at infinity in the second case). We prove that invariance of multiplicity in the local case is equivalent to invariance of degree in the global case. We prove invariance for curves and surfaces. In the way we prove invariance of the tangent cone and relative multiplicities at infinity under outer bi-Lipschitz homeomorphims at infinity, and that the abstract topol…

关于实 Milnor 数在渐近 Lipschitz 等价下的不变性

On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence

本文研究了函数芽在双 Lipschitz 及渐近 Lipschitz 等价下实 Milnor 数的不变性，引入 $α$-导数研究渐近增长，并确立了该数在 $C^k$ 和 $C^{\infty}$ 等价下保持不变的充分条件。

梯度峡谷上牛顿多边形的双Lipschitz不变性

Bi-Lipschitz invariance of Newton polygons along gradient canyons

通过极弧和梯度峡谷研究全纯函数芽的双Lipschitz右等价性，定义增广牛顿多边形并证明其在梯度峡谷上不变，揭示其紧边分解为拓扑部分和Lipschitz部分，后者编码第二级Henry-Parusiński型不变量。

Bi-Lipschitz invariance of the multiplicity

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is shown that $p$ is a singular point of the curve $C$ if and only if this multiplicity is greater than or equal to 2, in this sense, such an integer number can be considered as a measure of how singular can be a point of the curve $C$. In these notes, we address the classical concept of multiplicity of singular points of complex algebraic sets (not necessarily com…

双 Lipschitz 不变量在奇点理论中的应用：Lojasiewicz 指数和欧拉阻挠

Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

本文研究了奇点理论中两个基本局部不变量的双 Lipschitz 不变性，即 Lojasiewicz 指数和局部欧拉阻挠，探讨了其在双 Lipschitz 等价下的保持条件，并回答了局部欧拉阻挠是否为双 Lipschitz 不变量的问题。

关于次解析几何

On subanalytic geometry

本文是一篇次解析几何综述，梳理了全局次解析集的定义、基础工具、利普希茨几何进展及几何积分理论相关成果。