From H\"older Triangles to the Whole Plane

TL;DR

This paper presents a method to classify real polynomial functions based on Lipschitz equivalence by analyzing critical points and extends this approach to classify certain two-variable polynomials through reduction to single-variable problems.

Contribution

It introduces a new criterion for Lipschitz equivalence of single-variable polynomials and reduces the classification of specific two-variable polynomials to this simpler problem.

Findings

Lipschitz equivalence determined by critical points and multiplicities.

Reduction of two-variable polynomial classification to single-variable case.

Applicable to ${ m eta}$-quasihomogeneous polynomials under general conditions.

Abstract

We show how to determine whether two given real polynomial functions of a single variable are Lipschitz equivalent by comparing the values and also the multiplicities of the given polynomial functions at their critical points. Then we show how to reduce the problem of ${\cal R}$-semialgebraic Lipschitz classification of $\beta$-quasihomogeneous polynomials of two real variables to the problem of Lipschitz classification of real polynomial functions of a single variable, under some fairly general conditions.