Semialgebraic Lipschitz equivalence polynomial functions

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 $\beta$-transforms and inverse $\beta$-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 $({\mathbb{R}}^2,0)\to (\mathbb{R},0)$ admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli.