Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Raphael de Omena, José Edson Sampaio, Emanoel Souza

TL;DR

This work investigates when the real Milnor number $\mu(f)$ remains invariant under Lipschitz-type equivalences, introducing the $\alpha$-derivative to capture asymptotic growth. By defining and exploiting $\mathcal{R}$-asymptotically Lipschitz equivalence, the authors establish invariance of $\mu(f)$ under these conditions provided the initial parts $\mathrm{in}(f)$ and $\mathrm{in}(g)$ have algebraically isolated singularities, and they extend the framework to $C^k$ and $C^{\infty}$ equivalences. The results are complemented by demonstrations of sharpness via counterexamples and by corollaries for $C^1$ and $C^k$ trivial families, as well as invariance under diffeomorphisms between zero sets for irreducible real analytic germs. Overall, the paper broadens understanding of Milnor-number invariance beyond biholomorphic settings, unlocking new invariants under asymptotic and smooth equivalences with clear conditions and limitations.

Abstract

We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $α$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{\infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.

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

TL;DR

This work investigates when the real Milnor number remains invariant under Lipschitz-type equivalences, introducing the -derivative to capture asymptotic growth. By defining and exploiting -asymptotically Lipschitz equivalence, the authors establish invariance of under these conditions provided the initial parts and have algebraically isolated singularities, and they extend the framework to and equivalences. The results are complemented by demonstrations of sharpness via counterexamples and by corollaries for and trivial families, as well as invariance under diffeomorphisms between zero sets for irreducible real analytic germs. Overall, the paper broadens understanding of Milnor-number invariance beyond biholomorphic settings, unlocking new invariants under asymptotic and smooth equivalences with clear conditions and limitations.

Abstract

We investigate sufficient conditions for the invariance of the real Milnor number under -bi-Lipschitz equivalence for function-germs . More generally, we explore its invariance within the extended framework of -asymptotically Lipschitz equivalence. To this end, we introduce the -derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of and equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.
Paper Structure (5 sections, 14 theorems, 35 equations)

This paper contains 5 sections, 14 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic Lipschitz Invariants of Real Mappings
  4. Invariance of the Milnor Number under Lipschitz equivalences
  5. Invariance of the Milnor Number under $C^k$ equivalences

Key Result

Theorem \ref{Cor:Invariance}

Let $f, g\colon (\mathbb{R}^n,0)\to (\mathbb{R},0)$ be germs of $C^{\infty}$ functions. Suppose that ${\rm in}(f)$ and ${\rm in}(g)$ have algebraically isolated singularities at the origin. If $f$ and $g$ are $\mathcal{R}$-asymptotically Lipschitz equivalent at the origin, then $\mu(f) = \mu(g)$.

Theorems & Definitions (36)

  • Example 1.1: Wall:1983
  • Example 1.2
  • Theorem \ref{Cor:Invariance}
  • Theorem \ref{Teo:Cinfinito}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 3.1
  • ...and 26 more