BILIPSCHITZ INVARIANCE OF THE MULTIPLICITY

Abstract

A famous problem of Zariski, stated in 1971 [21], is to decide whether the classical algebro-geometric multiplicity m_o(f) of a complex hypersurface f⁻¹(0) at an isolated singularity O is an invariant of the topological type of the embedded germ. That is, if f[ratio ]Cⁿ, O→C, 0 and g[ratio ]Cⁿ, O→C, 0 are two germs of polynomial functions, and there is a homeomorphism ϕ defined between two open neighbourhoods U and V of O in Cⁿ such that ϕ(U) = V, ϕ(U∩f⁻¹(0)) = V∩g⁻¹(0) and ϕ(O) = O, then is it always the case that m_o(f) = m_o(g)? Here the multiplicity m_o(f) is defined to be the intersection number of f⁻¹(0) with a generic complex line L passing through O in Cⁿ, that is, the number of points of intersection of f⁻¹(0) with a line which is an arbitrarily small perturbation of L. If f is reduced, then m_o(f) is equal to the order of the polynomial f(z) at O.