Bi-Lipschitz invariance of Newton polygons along gradient canyons

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.