曲面的 Łojasiewicz 指数：一种内蕴视角

Lojasiewicz exponents and resolution of singularities

We show an effective method to compute the Łojasiewicz exponent of an arbitrary sheaf of ideals of $\OO_X$, where $X$ is a non-singular scheme. This method is based on the algorithm of resolution of singularities.

代数Lojasiewicz指数的理论

An algebraic theory of Lojasiewicz exponents

本文提出一种统一的代数与赋值理论，研究Lojasiewicz指数的渐近包含阈值，通过有限最大原理将无限赋值上界转化为有限最大值，揭示指数的本质是赋值优化问题，并在族中建立稳定性与分层现象。

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。

某一类光滑函数中的Łojasiewicz不等式

Lojasiewicz inequalities in a certain class of smooth functions

本文研究满足Kouchnirenko非退化与Kamimoto–Nose条件的光滑函数芽，证明其满足Łojasiewicz不等式，且凸情形下的所有Łojasiewicz指数可由Newton多面体简洁表示。

On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Lojasiewicz exponent of an analytic function is equal to one half at a critical point, then the function is Morse-Bott and thus its critical set nearby is an analytic submanifold. The main ingredients in our proofs are the Lojasiewicz gradient inequality for an analytic function on a…

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $Ø_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the Łojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.