Bi-Lipschitz invariance of Newton polygons along gradient canyons

TL;DR

This paper demonstrates that augmented Newton polygons associated with polar arcs and gradient canyons are invariant under bi-Lipschitz right-equivalence, providing new invariants for classifying holomorphic function germs.

Contribution

It introduces the invariance of augmented Newton polygons along gradient canyons under bi-Lipschitz equivalence and defines new discrete invariants like polar multiplicity.

Findings

Augmented Newton polygons are constant along gradient canyons.

The compact edges decompose into topological and Lipschitz parts.

Polar multiplicity equals the horizontal length of the top edge.

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 $\gamma$ we consider the Newton polygon of $f_x(X+\gamma(Y),Y)$ and define its augmentation by adjoining the point $(0,\text{ord } f(\gamma(y),y)-1)$. 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-Parusi\'nski 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.