Resumen de Metric theory of semialgebraic curves

Lev Birbrair, Alexandre C. G. Fernandes

• 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.