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.
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BibTeX
@misc{nguyen2026bilipschitz,
title = {{Bi-Lipschitz invariance of the transverse polynomial along polar arcs}},
author = {Nhan Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06005},
eprint = {2610.06005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06005},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Bi-Lipschitz invariance of the transverse polynomial along polar arcs
AU - Nhan Nguyen
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06005
UR - https://arxiv.org/abs/2610.06005
AB - 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.
ER -