bi-Lipschitz versus Analytic equivalence of two variable complex quasihomogeneous function-germs

TL;DR

The paper proves a rigidity phenomenon for complex quasihomogeneous polynomials in two variables: two non-homogeneous polynomials with the same weights $(p,q)$ are right bi-Lipschitz equivalent at $0$ if and only if they are analytically equivalent as function-germs in $\mathbb{C}^2$. Central tools include height functions, a $\varpi$-weighted blow-up construction, and a reduction to one-variable automorphism classifications, which together force compatibility of zeros and multiplicities under bi-Lipschitz maps. The authors show that the height function is preserved up to automorphisms of $\mathbb{C}$ under bi-Lipschitz equivalence and derive explicit coordinate changes in the main cases ($p=1$ and $p>1$) to establish analytic equivalence from bi-Lipschitz data. This work strengthens known rigidity results and clarifies the relationship between Lipschitz and analytic categories for two-variable quasihomogeneous singularities.

Abstract

In this paper we address the problem of classifying complex (non-homogeneous) quasihomogeneous polynomials in two variables under bi-Lipschitz equivalence. We prove that pairs of such polynomials are (right) bi-Lipschitz equivalent as function-germs at $0\in\mathbb{C}^{2}$ iff they are analytically equivalent.

Theorems & Definitions (20)

• Example 1.1: 1965, H. Whitney
• Theorem 1.2
• Proposition 3.1
• Lemma 3.2
• proof
• proof : Proof of Proposition \ref{['prop:one_variable']}
• Proposition 3.3
• Remark 4.1
• Proposition 5.1
• proof