Metric theory of semialgebraic curves

• Autores: Lev Birbrair, Alexandre C. G. Fernandes
• Localización: Revista matemática complutense, ISSN-e 1988-2807, ISSN 1139-1138, Vol. 13, Nº 2, 2000, págs. 369-382
• Idioma: inglés
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• Resumen

  • We present a complete bi-Lipschitz classification of germs of semialgebraic curves (semialgebraic sets of the dimension one). For this purpose we introduce the so-called Hölder Semicomplex, a bi-Lipschitz invariant. Hölder Semicomplex is the collection of all first exponents of Newton-Puiseux expansions, for all pairs of branches of a curve. We prove that two germs of curves are bi-Lipschitz equivalent if and only if the corresponding Hölder Semicomplexes are isomorphic. We also prove that any Hölder Semicomplex can be realized as a germ of some plane semialgebraic curve. Finally, we compare these Hölder Semicomplexes with Hölder Complexes-complete bi-Lipschitz invariant of two-dimensional semialgebraic sets.