Outer Lipschitz Geometry of Unions of Tangential Nodal Sets

Source 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 $(X,x_0)$ 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.