Zariski multiplicity conjecture

An equation can describe a shape near a point where it is singular. For example, x²+y³+z⁵=0 starts with a degree-two term; this first degree is its multiplicity. If two such local shapes can be matched by continuously reshaping the surrounding space, must their multiplicities agree? The equations must have no repeated factors; the open target concerns complex spaces with at least three coordinates.[2]

A local equation contains degree information that a continuous reshaping need not visibly preserve. Must the topology of a complex hypersurface together with its surrounding space retain that first degree? For example, $f=x^2+y^3+z^5$ has first nonzero homogeneous term $x^2$, so its multiplicity is two; this example does not establish an equivalence with any other singularity.

Precisely, for reduced holomorphic germs $f,g:(\mathbb C^n,0)\to(\mathbb C,0)$, $n\ge3$, does an ambient homeomorphism carrying $V(f)$ to $V(g)$ imply $\operatorname{ord}_0 f=\operatorname{ord}_0 g$? Reduced excludes repeated analytic factors, and a germ records only a sufficiently small neighbourhood of the point.[2]

The answer is known for plane curves and for certain continuous families with isolated singularities. A stronger kind of matching that controls distance distortion also preserves multiplicity two. Those results leave the question for arbitrary pairs of higher-dimensional shapes under continuous matching alone.[1] [2] [3]

The known constant-multiplicity theorem for topologically constant families assumes isolated singularities and a continuous family; it is not a theorem comparing every pair of germs. The metric multiplicity-two result assumes ambient bi-Lipschitz equivalence, which controls distance distortion. Neither restriction can be removed merely by citing those results.[1] [2] [3]

Equations and continuous shape give different descriptions of a singularity. If shape determines multiplicity, two equations with different first degrees cannot describe the same embedded local topology. This would connect an easy equation-based distinction to the harder task of classifying singular shapes under continuous changes.[2]

Multiplicity helps distinguish analytic singularities, while embedded topology records what survives continuous changes of coordinates. Relating the two would make a numerical feature of the equation available to topological classification. The family results show why this matters for equisingularity: a deformation can change coefficients while retaining the local topological type, and one wants to know which analytic data are then forced to stay constant.[1] [2]