Counterexamples to Zariski’s multiplicity conjecture

The explanation

Take the polynomial h = T(X − T²)(X − 2T²) in two variables. Multiply it out and you get TX² − 3XT³ + 2T⁵. The terms have total degrees 3, 4 and 5. The lowest is 3, so the multiplicity of h at the origin is 3. The multiplicity is the least total degree of a term that actually appears. It counts how many sheets of the shape h = 0 meet at the origin. Here the three factors give three smooth sheets. The second paper starts its construction from exactly this curveSource 1.

Now look at the same shape with topology instead. A homeomorphism is a continuous deformation with a continuous inverse. It may stretch and bend, but it may not tear. A germ is a shape seen only near the origin. Two germs are ambiently homeomorphic when a deformation of the surrounding space carries one onto the other. Zariski asked in 1971 whether this embedded topology always determines the multiplicitySource 2.

For curves in the plane, the answer is yesSource 3. In every dimension, a germ that is ambiently homeomorphic to a smooth one is itself smooth. That is the A’Campo–Lê theoremSource 4. So multiplicity one can never pair with a larger multiplicitySource 4. Many other partial results needed extra assumptions. Some kept a continuous family with a fixed Milnor number, a count of the holes in a nearby smooth sliceSource 5. Others assumed metric or contact structure, or few variablesSource 6.

The first paper claims a negative answer. It builds two polynomial germs in N complex variables, where N is a multiple of eight and larger than 3Source 7. One has multiplicity 2 and the other has multiplicity 3Source 8. Both have an isolated critical point at the origin, meaning all partial derivatives vanish together only thereSource 7. Yet the two zero sets are ambiently homeomorphicSource 8. The paper fixes N by exact arithmetic but does not try to make it smallSource 9. The same pair also answers Arnold’s corank question negatively for ambient topologySource 10. The corank is the number of variables minus the rank of the matrix of second derivatives.

The second paper brings the dimension down to four variables. It builds two germs in four complex variables with isolated critical pointsSource 11. One has multiplicity 4 and the other multiplicity 5. Their zero sets are again ambiently homeomorphicSource 11. The paper says its argument is independent of the first, with a different dimension and equation formatSource 12. The case of three variables remains open in both papersSource 12.

Why was this hard? Topology is usually read from the Milnor fibre, a smooth nearby slice of the shape. Its holes carry an integer pairing called the Seifert form, which records how cycles link. In high dimensions this form decides the embedded topology. Multiplicity, by contrast, is read directly from the lowest-degree terms of an equationSource 13. The main difficulty is to keep the two different lowest degrees while making the integer Seifert forms match exactlySource 14. Matching rational spectra is not enough. The integer forms must be identifiedSource 14.

Families cannot do this. The theorem of Fernández de Bobadilla and Pełka says multiplicity is constant along continuous families with constant Milnor number. So these germs cannot be joined by such a familySource 15.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not settle Zariski’s question in three complex variables, and it does not address families of germs.