Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity?

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The Budur-Fernandez de Bobadilla-Le-Nguyen conjecture on initial-form Milnor fibres of topologically equivalent germs evidence for this event

Verification scripts for the finite arithmetic (September 24 paper) evidence for this event

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Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity? parent of this event

Companion: Ambiently homeomorphic isolated hypersurface germs in C^4 with multiplicities four and five evidence for this event

VibeMathed record: Zariski's multiplicity conjecture: does embedded topology determine the multiplicity of a hypersurface singularity? evidence for this event

Zariski, Some open questions in the theory of singularities, Bull. AMS 77 (1971) evidence for this event

Two independent counterexample constructions. September 24: for some finite $N>3$ divisible by eight, reduced weighted homogeneous real polynomials $f_1,f_2$ on $\mathbb{C}^N$ with isolated critical points, $\mathrm{ord}_0f_1=2$, $\mathrm{ord}_0f_2=3$, and $(\mathbb{C}^N,V(f_1),0)\cong(\mathbb{C}^N,V(f_2),0)$; the germs are also topologically right equivalent, so the right-equivalence version and its mod-2 version (Sampaio 2025, Conjectures 2 and Z mod 2) fail too. September 27: reduced convergent germs $f_4,f_5$ on $\mathbb{C}^4$ with isolated critical points, orders $4$ and $5$, ambiently homeomorphic zero sets. Not addressed: hypersurfaces in $\mathbb{C}^3$, and metric (bi-Lipschitz) or family versions, where positive results stand. parent of this event