A famous problem of Zariski, stated in 1971 [21],
is to decide whether the classical
algebro-geometric multiplicity mo(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 mo(f) =
mo(g)? Here the multiplicity
mo(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 mo(f) is equal
to the
order of the polynomial f(z) at O.