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Metric Zariski Multiplicity Conjecture

Establish that for any two complex analytic hypersurface germs (X,0) and (Y,0) in (C^n,0) that are ambient bi-Lipschitz homeomorphic, the multiplicities m(X,0) and m(Y,0) are equal.

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Background

The metric version of Zariski's classical multiplicity problem asks whether multiplicity is preserved under ambient bi-Lipschitz homeomorphisms of hypersurface germs in complex Euclidean space. The authors prove this conjecture for the special case of multiplicity two and relate its resolution to a metric version of Zariski's problem on cones (Conjecture B).

They also show that ambient homeomorphic algebraic cones have bases with the same Euler characteristic, giving additional invariants toward the conjecture. Despite these advances, the general conjecture remains unresolved.

References

Conjecture A. { i Let $X,Y\subset Cn$ be two complex analytic hypersurfaces . If their germs at zero are ambient bi-Lipschitz homeomorphic, then their multiplicities $m(X,0)$ and $m(Y,0)$ are equal.} This conjecture is still open.

— Metric version of the Zariski Multiplicity Conjecture is true for multiplicity two (2509.03447 - Fernandes et al., 3 Sep 2025) in Section 1 (Introduction), Conjecture A