Lipschitz Geometry of Mixed Polynomials

TL;DR

This paper explores the bi-Lipschitz geometry of two-variable mixed polynomials, establishing conditions for triviality, defining metric invariants, and advancing classification methods beyond traditional invariants like Newton boundaries.

Contribution

It introduces new invariants based on face diagrams for bi-Lipschitz classification of mixed polynomials, extending understanding beyond Newton boundary invariants.

Findings

Ambient bi-Lipschitz V-triviality holds under specific weighted homogeneity conditions.

Two simple metric links suffice for classification within certain polynomial classes.

Newton boundary and C-face diagram are not invariants of bi-Lipschitz V-equivalence.

Abstract

We investigate the (ambient) bi-Lipschitz V-equivalence of two-variable mixed polynomials satisfying the Newton inner non-degeneracy condition. Concerning triviality, we show that ambient bi-Lipschitz V-triviality for families $\{f + \varepsilon \theta\}_{\varepsilon \in \mathbb{R}}$ is guaranteed when $f$ is semi-radially weighted homogeneous and the weighted radial degree of every monomial in $\theta$ is greater than the weighted radial degree associated with $f$. However, in the general case, we prove that it is not guaranteed, even though ambient topological V-triviality still holds. For the classification problem, we define two simple metric links and prove that they suffice to determine bi-Lipschitz V-equivalence within the class of mixed polynomials that are $\Gamma_{\rm inn}$-nice. A key outcome is that neither the Newton boundary $\Gamma(f)$ nor the C-face diagram $\Gamma_{\rm inn}$ constitutes an invariant of this equivalence for such mixed polynomials. To outcome this, we introduce new data extracted from the two face diagrams under consideration and prove that, under certain generic conditions, these data become fundamental invariants for the bi-Lipschitz equivalences. This provides a fundamental step toward a bi-Lipschitz classification of these mixed polynomials.