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Hölder invariance of the Henry-Parusinski invariant

Alexandre Fernandes, José Edson Sampaio, Joserlan Perote da Silva

Abstract

In this article, we show the Hölder invariance of the Henry-Parusinski invariant. For a single germ $ f$, the Henry-Parusinski invariant of $ f $ is given in terms of the leading coefficients of the asymptotic expansion of $ f $ along the branches of the generic polar curve of $f$. As a consequence, we obtain that the classification problem of polynomial function-germs, with uniformly bounded degree, under Hölder equivalence, admits continuous moduli.

Hölder invariance of the Henry-Parusinski invariant

Abstract

In this article, we show the Hölder invariance of the Henry-Parusinski invariant. For a single germ , the Henry-Parusinski invariant of is given in terms of the leading coefficients of the asymptotic expansion of along the branches of the generic polar curve of . As a consequence, we obtain that the classification problem of polynomial function-germs, with uniformly bounded degree, under Hölder equivalence, admits continuous moduli.
Paper Structure (10 sections, 14 theorems, 75 equations)

This paper contains 10 sections, 14 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz and Hölder mappings.
  4. Other distances
  5. Hölder invariants
  6. Hölder invariance of the multiplicity of functions
  7. Special subsets preserved under bi-$\alpha$-Hölder conjugations
  8. The Henry-Parunsinski Invariant
  9. Cuspidal Neighborhoods of Polar Curves
  10. An application: Existence of continuous moduli

Key Result

Theorem 2.5

Let $X\subset\mathbb{R}^N$ be a path-connected subset. Assume that $X$ is a definable set in an o-minimal structure on $\mathbb{R}$. Then $d_{X, diam}$ is equivalent to $d_X$.

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • ...and 21 more