Counterexamples to Zariski’s multiplicity conjecture

What OpenAI claims

Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.

Papers (2)

1. Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.

2. Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five

   27 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.