Lipschitz Geometry of Mixed Polynomials

TL;DR

The work investigates ambient bi-Lipschitz V-equivalence for two-variable mixed polynomials under inner non-degeneracy, showing semi-radial cases yield ambient V-triviality under degree constraints while the general case can fail topologically. It introduces two simple metric links and a refined analysis via Γ_inn-face data to advance a Lipschitz classification within the Γ_inn-nice setting, revealing that Newton boundary and Γ_inn diagrams need not be invariant. By building obstruction-aware horn decompositions, test-arcs, and tangent-cone comparisons, the authors derive both necessary and (under extra hypotheses) sufficient invariants for bi-Lipschitz equivalence, including the Type I–III taxonomy and the contact data NC(f). The results illuminate how Lipschitz geometry of mixed polynomials can be rigid in surprising ways, even when topological types remain unchanged, and they lay groundwork for a broader bi-Lipschitz classification program in this non-holomorphic setting.

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 $\{f + \varepsilon θ\}_{\varepsilon \in \mathbb{R}}$ is guaranteed when $f$ is semi-radially weighted homogeneous and the weighted radial degree of every monomial in $θ$ 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 $Γ_{\rm inn}$-nice. A key outcome is that neither the Newton boundary $Γ(f)$ nor the C-face diagram $Γ_{\rm inn}$ 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.

Figures (6)

• Figure 1: The Newton polygon $\Gamma_+(f)$ in Example \ref{['exemplo-poligono-newton']}.
• Figure 2: The $C$-diagram $\Gamma$ in Example \ref{['exemplo-poligono-newton']}.
• Figure 3: Nested link. Notice that $[*]$ represents the components in $\mathbb{C}\times \mathbb{S}^1$ and $[*]'$ represents the components in $\mathbb{S}^1\times \mathbb{C}$
• Figure 4:
• Figure 5:

Theorems & Definitions (134)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6