The Zariski multiplicity conjecture asks whether the embedded topological germ of a reduced complex hypersurface determines its
multiplicity. For a nonzero
convergent power series 
vanishing at the origin,
the multiplicity is 
, the least total degree
of a nonzero monomial. Reduced
means that no irreducible factor is repeated. The topological equivalence is a homeomorphism germ of ambient
pairs, fixing the origin and carrying one zero
set onto the other.
The metric version restricts the ambient equivalence to a bi-Lipschitz map. Sampaio (2026, Theorem 4.1) proves that the multiplicity
is preserved for reduced holomorphic hypersurface
germs whenever each irreducible component
of the tangent cone has singular locus of complex
dimension at most two. In particular, this proves the
metric assertion in ambient spaces 
for 
.
The topological conjecture permits general homeomorphisms
and is a different assertion. Sampaio (2026) reports AI assistance in refining the
mathematical arguments and states that the suggestions were independently checked.
