Metric version of the Zariski Multiplicity Conjecture is true for multiplicity two

Abstract: We prove that if two algebraic $(n-1)$-dimensional cones $P, R\subset\mathbb C^n$ are ambient homeomorphic, then their bases $B_P$ and $B_R$ have the same Euler characteristic. As an application, we show our main result: if $(X,0),(Y,0)\subset (\C^n,0)$ are germs of analytic hypersurfaces, which are ambient bi-Lipschitz equivalent and $m_0(X)=2$, then also $m_0(Y)=2.$. As another application, we also obtain a complete positive answer to the Metric Arnold-Vassiliev Problem, which says that if two holomorphic functions have zeros that are ambient bi-Lipschitz invariant, then the Hessian matrices of the functions have the same rank.