Bi-Lipschitz invariance of the transverse polynomial along polar arcs

Abstract

We study transverse polynomials along polar arcs of reduced holomorphic plane function germs. We prove that, under bi-Lipschitz right equivalence, matched tangential polar arcs have the same transverse order and their transverse polynomials agree up to nonzero rescalings of the source and target, at every rational scale $1<q<d(γ)$, where $d(γ)$ is the gradient canyon degree. This range is sharp as the result can fail at both endpoints, even under analytic right equivalence. As an application, we recover the bi-Lipschitz invariance of the augmented Newton polygon proved by Migus--Păunescu--Tibăr. We also give an example showing that the transverse polynomial contains new information not detected by the polar-value invariants and augmented Newton polygons.