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Semialgebraic Lipschitz equivalence polynomial functions

Sergio Alvarez

TL;DR

This work establishes a framework to classify real $\beta$-quasihomogeneous polynomials in two variables under semialgebraic bi-Lipschitz equivalence by reducing the problem to the Lipschitz classification of associated height functions in one variable. Central to the approach are the $\beta$-transform and inverse $\beta$-transform, together with the zygothetic group acting on height-function pairs, which connect 2D equivalence to 1D data and provide structural conditions ensuring equivalence. The authors prove criteria under which height-function Lipschitz equivalence implies $F$ and $G$ are ${\cal R}$-semialgebraically Lipschitz equivalent, and they develop a theory of $\beta$-regular zygotheties to guarantee the construction of inverse transforms. The Henry–Parusiński example is revisited to demonstrate that continuous moduli persist in the semialgebraic Lipschitz setting, showing that real two-variable quasihomogeneous polynomials can exhibit rich moduli behavior similar to analytic germs, now within a semialgebraic framework.

Abstract

We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in ${\mathbb R}^2$. Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single-variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of $β$-transforms and inverse $β$-transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusiński to show that the bi-Lipschitz equivalence of analytic function germs $({\mathbb R}^2,0)\rightarrow({\mathbb R},0)$ admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli.

Semialgebraic Lipschitz equivalence polynomial functions

TL;DR

This work establishes a framework to classify real -quasihomogeneous polynomials in two variables under semialgebraic bi-Lipschitz equivalence by reducing the problem to the Lipschitz classification of associated height functions in one variable. Central to the approach are the -transform and inverse -transform, together with the zygothetic group acting on height-function pairs, which connect 2D equivalence to 1D data and provide structural conditions ensuring equivalence. The authors prove criteria under which height-function Lipschitz equivalence implies and are -semialgebraically Lipschitz equivalent, and they develop a theory of -regular zygotheties to guarantee the construction of inverse transforms. The Henry–Parusiński example is revisited to demonstrate that continuous moduli persist in the semialgebraic Lipschitz setting, showing that real two-variable quasihomogeneous polynomials can exhibit rich moduli behavior similar to analytic germs, now within a semialgebraic framework.

Abstract

We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in . Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single-variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of -transforms and inverse -transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusiński to show that the bi-Lipschitz equivalence of analytic function germs admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli.

Paper Structure

This paper contains 10 sections, 37 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Lipschitz equivalence of polynomial functions of a single variable
  3. Lipschitz classification of polynomial functions of a single variable
  4. On the bi-Lipschitz transformation $\phi$
  5. ${\cal R}$-Semialgebraic Lipschitz equivalence of $\beta$-qua-si-ho-mo-ge-ne-ous polynomials
  6. $\beta$-regular germs and the $\beta$-transform
  7. The zygothetic group
  8. $\beta$-regular zygotheties and the inverse $\beta$-transform
  9. Some conditions for replacing zygotheties with $\beta$-regular zygotheties
  10. Henry and Parusiński's example revisited

Key Result

Lemma 2.1

Let $f,g\colon\mathbb R\to\mathbb R$ be nonconstant polynomial functions. If $\phi\colon\mathbb R\to\mathbb R$ is a homeomorphism such that $g\circ\phi = f$, then $\phi$ is semialgebraic.

Theorems & Definitions (88)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.2
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Theorem 2.1a
  • ...and 78 more