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Bi-Lipschitz invariance of Newton polygons along gradient canyons

Piotr Migus, Laurenţiu Păunescu, Mihai Tibăr

TL;DR

The paper addresses identifying analytic features of holomorphic germs in two variables preserved by bi-Lipschitz right-equivalence by using polar arcs and gradient canyons. It introduces the augmented Newton polygon $\widehat{\mathcal{NP}}(f_x,\mathcal{C})$ along a canyon and proves its bi-Lipschitz invariance for canyon degree $d_{\mathcal{C}}>1$, along with a decomposition into topological and Lipschitz parts; the Lipschitz part encodes second-level Henry-Parusiński invariants and the top edge yields the polar multiplicity $\mu_f(\mathcal{C})$, a new discrete invariant. The framework connects Newton polygon data to the Kuo-Lu tree and Puiseux arcs, providing a concrete finite invariant set attached to each gradient canyon. Examples illustrate invariance while demonstrating non-completeness and highlight how second-level invariants refine the bi-Lipschitz classification beyond first-level data.

Abstract

We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $γ$ we consider the Newton polygon of $f_x(X+γ(Y),Y)$ and define its augmentation by adjoining the point $(0,\text{ord } f(γ(y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusiński type invariants. As an application we introduce the polar multiplicity of a canyon and identify it with the horizontal length of the top edge of the augmented polygon, yielding a new discrete bi-Lipschitz invariant.

Bi-Lipschitz invariance of Newton polygons along gradient canyons

TL;DR

The paper addresses identifying analytic features of holomorphic germs in two variables preserved by bi-Lipschitz right-equivalence by using polar arcs and gradient canyons. It introduces the augmented Newton polygon along a canyon and proves its bi-Lipschitz invariance for canyon degree , along with a decomposition into topological and Lipschitz parts; the Lipschitz part encodes second-level Henry-Parusiński invariants and the top edge yields the polar multiplicity , a new discrete invariant. The framework connects Newton polygon data to the Kuo-Lu tree and Puiseux arcs, providing a concrete finite invariant set attached to each gradient canyon. Examples illustrate invariance while demonstrating non-completeness and highlight how second-level invariants refine the bi-Lipschitz classification beyond first-level data.

Abstract

We study bi-Lipschitz right-equivalence of holomorphic function germs via polar arcs and gradient canyons. For a polar arc we consider the Newton polygon of and define its augmentation by adjoining the point . We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusiński type invariants. As an application we introduce the polar multiplicity of a canyon and identify it with the horizontal length of the top edge of the augmented polygon, yielding a new discrete bi-Lipschitz invariant.
Paper Structure (15 sections, 8 theorems, 71 equations, 6 figures)

This paper contains 15 sections, 8 theorems, 71 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gradient degree and gradient canyon.
  4. The order $\mathop{\rm{ord}} f(\gamma(y),y)$.
  5. Clusters of canyons.
  6. Contact of canyons.
  7. Sub-clusters of canyons.
  8. Lipschitz invariants and "identity card".
  9. Higher invariants: second level
  10. Newton polygon and related results
  11. Gradient degree
  12. Auxiliary lemma
  13. The impact of the bi-Lipschitz equivalence on the Newton polygon $\mathcal{NP}(f_x,\gamma)$
  14. Multiplicity of gradient canyons
  15. Examples

Key Result

Theorem 2.2

Let $f,g :({\mathbb C}^2,0)\to ({\mathbb C},0)$ be holomorphic functions such that $f=g\circ \varphi$, where $\varphi:({\mathbb C}^2,0)\to ({\mathbb C}^2,0)$ is a bi-Lipschitz homeomorphism and $f$ is mini-regular in $x$. Let $h=\mathop{\rm{ord}} f(\gamma(y),y) = \mathop{\rm{ord}} f(\gamma'(y),y)$. then the effect of the bi-Lipschitz map $\varphi$ on the pair $\bigl( H, (\tilde{a}-\tilde{a}')\big

Figures (6)

  • Figure 1: Construction of the line $L$ whose co-slope equals the gradient degree.
  • Figure 2: Extension of $E_\delta$ to the vertical axis in $\mathcal{NP}(f_x,\gamma)$.
  • Figure 3: For $t\neq 0$: the solid line segments are $\mathcal{NP}((f_t)_x,\gamma)$ for a representative polar arc $\gamma$ in the canyon, and the dashed segment is the extra edge in $\widehat{\mathcal{NP}}((f_t)_x,{\mathcal{C}})$ coming from the convex hull with $(0,h-1)$.
  • Figure 4: For $t=0$: the same convention as in Figure \ref{['fig:ft-canyon-tnz']}.
  • Figure 5: For $t\neq 0,1$: solid line is $\mathcal{NP}((f_t)_x,\gamma)$ for a representative polar arc $\gamma$ in the canyon, and dashed line is the extra edge in $\widehat{\mathcal{NP}}((f_t)_x,{\mathcal{C}})$ coming from the convex hull with $(0,h-1)$.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 4.2: Well-definedness on a gradient canyon
  • Lemma 4.3
  • proof
  • Corollary 4.4
  • proof
  • Theorem 4.5
  • proof
  • ...and 4 more