Outer Lipschitz Geometry of Unions of Tangential Nodal Sets
Abstract
In 2000, Birbrair and Fernandes proved that the Hölder Semicomplex of the branches of a germ of a curve with its contact orders is a complete invariant of outer bi-Lipschitz equivalence of definable sets in polynomially bounded o-minimal structures. They also proved a realization theorem for it, i.e., for each Hölder semicomplex G, there is a germ of a semialgebraic curve (X,x0) such that the Hölder Semicomplex of (X,x0) coincides with G. In 2000, Birbrair proposed the problem of finding and characterizing all the Hölder semicomplexes that can be realized as a Hölder Semicomplex of a real analytic curve. Solving this problem in its full generality is a notoriously hard task; however, in this paper, we provide a complete answer to this problem when we restrict it to tangential nodal curves: curves defined as the union of the zero set of a harmonic function and its tangent cone at a fixed point.
Related
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BibTeX
@misc{sampaio2026outer,
title = {{Outer Lipschitz Geometry of Unions of Tangential Nodal Sets}},
author = {José Edson Sampaio and Luan Moura Sousa and Emanoel Ferreira de Souza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36381},
eprint = {2610.36381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36381},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Outer Lipschitz Geometry of Unions of Tangential Nodal Sets
AU - José Edson Sampaio
AU - Luan Moura Sousa
AU - Emanoel Ferreira de Souza
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36381
UR - https://arxiv.org/abs/2610.36381
AB - In 2000, Birbrair and Fernandes proved that the Hölder Semicomplex of the branches of a germ of a curve with its contact orders is a complete invariant of outer bi-Lipschitz equivalence of definable sets in polynomially bounded o-minimal structures. They also proved a realization theorem for it, i.e., for each Hölder semicomplex G, there is a germ of a semialgebraic curve (X,x_0) such that the Hölder Semicomplex of (X,x_0) coincides with G. In 2000, Birbrair proposed the problem of finding and characterizing all the Hölder semicomplexes that can be realized as a Hölder Semicomplex of a real analytic curve. Solving this problem in its full generality is a notoriously hard task; however, in this paper, we provide a complete answer to this problem when we restrict it to tangential nodal curves: curves defined as the union of the zero set of a harmonic function and its tangent cone at a fixed point.
ER -