Multiplicity mod 2 as a Metric Invariant
We study the multiplicity modulo 2 of real analytic hypersurfaces. We prove that, under some assumptions on the singularity, the multiplicity modulo 2 is preserved by subanalytic bi-Lipschitz homeomorphisms of a"e (n) . In the first part of the paper, we find a subset of the tangent cone which determines the multiplicity mod 2 and prove that this subset of S (n) is preserved by the antipodal map. The study of such subsets of S (n) enables us to deduce the subanalytic metric invariance of the multiplicity modulo 2 under some extra assumptions on the tangent cone. We also prove a real version of a theorem of Comte, and yield that the multiplicity modulo 2 is preserved by arc-analytic bi-Lipschitz homeomorphisms.
multiplicity metric invariant real analytic hypersurfaces