1 · The essentials
Can the shape of a complex equation’s solution set determine how strongly the equation vanishes at a special point? The manuscripts address this question by comparing shapes only near that point. They claim examples whose surrounding spaces can be continuously rearranged to carry one shape onto the other, yet whose orders of vanishing differ. In the main manuscript, those orders are two and three, and each shape has just one singular point nearby. A companion manuscript claims examples in four complex coordinates with orders four and five. These constructions are presented as negative answers to Zariski’s multiplicity question. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine two patterns on flexible sheets that can be matched by stretching the sheets without tearing. The manuscripts claim that such a match can preserve the surrounding shape while failing to preserve a numerical feature of the equations that describe it; the sheet image does not capture the complex coordinates or the local nature of the question.
Why it matters
The claimed examples separate two ways of describing a hypersurface: its shape within the surrounding space and the multiplicity of its defining equation. According to the main abstract, the same examples also give a negative answer to Arnold’s corank problem for ambient topological equivalence. The companion manuscript places a multiplicity counterexample specifically in four complex variables.
2 · Going further
The embedded Zariski multiplicity question asks whether ambiently homeomorphic reduced holomorphic hypersurface germs must have the same multiplicity. A germ records a space or function only near a specified point, and ambient homeomorphism requires a homeomorphism of the surrounding space carrying one zero-set germ onto the other. For a defining holomorphic function, multiplicity is its order of vanishing at that point. The reduced hypothesis excludes repeated factors in the defining equation.
The main manuscript claims two such hypersurface germs with multiplicities and . Both have isolated singularities and lie in a common complex affine space whose dimension is divisible by . The abstract further states that their defining function germs are topologically right equivalent: a homeomorphic change of coordinates identifies the functions themselves, not merely their zero sets. It also states that these examples answer Arnold’s corank problem negatively for ambient topological equivalence.
The companion manuscript claims reduced convergent holomorphic function germs in , with isolated critical points and multiplicities and , whose zero-set germs are ambiently homeomorphic. The shared approach described in the abstracts is construction of pairs with matching ambient topology but unequal multiplicities. The abstracts do not explain how the pairs or the homeomorphisms are constructed, so no further mechanism of the argument can be supplied from this source.
Useful notions
- Complex coordinates and holomorphic functions
- Zero sets of equations
- Local study near a point
- Homeomorphisms
- Order of vanishing
What was known before / what is new
Before
The source identifies Zariski’s embedded multiplicity conjecture as the question being answered negatively. It also names Arnold’s corank problem as a related question addressed by the main examples. It supplies no earlier partial results or historical account.
New
The main manuscript claims ambiently homeomorphic reduced hypersurface germs with isolated singularities and multiplicities two and three, with topological right equivalence of their defining functions as well. The companion claims ambiently homeomorphic zero-set germs in four complex variables arising from reduced convergent holomorphic functions with isolated critical points and multiplicities four and five.
Limits and assumptions
These are local claims about reduced holomorphic hypersurfaces, not statements about arbitrary global spaces. The main examples have isolated singularities in a common complex affine space of dimension divisible by eight; the companion examples involve isolated critical points in four complex variables. Topological right equivalence is stated for the main examples but not in the companion abstract. The supplied abstracts omit construction details and do not identify which further variants remain open. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Hypersurface germ —
- The local zero set of one holomorphic equation, considered only near a specified point.
- Multiplicity —
- The order of vanishing of a defining function at the point under study.
- Reduced —
- Having no repeated factors in the local defining equation.
- Ambient homeomorphism —
- A continuous, invertible rearrangement of the surrounding space, with continuous inverse, that carries one specified zero set onto the other.
- Isolated singularity —
- A singular point with no other singular points sufficiently nearby.
- Topological right equivalence —
- An identification of two function germs by a homeomorphic change of coordinates in their domain.
Keywords : Zariski multiplicity conjecture · Hypersurface germs · Ambient homeomorphism
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
24/09/2026We 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.
Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five
27/09/2026We 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.