Family 059 — The same local topology with different multiplicities

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Family 059 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Near a singular point, the lowest-degree term of an equation is an elementary measure of its complexity. A quadratic term gives multiplicity two; if all quadratic terms vanish and a cubic term remains, the multiplicity is three. Can the shape of the zero set, including how it sits in its ambient space, recover that number?

Zariski’s multiplicity question asks whether the answer is yes under an ambient homeomorphism. Such a map may bend the space continuously but cannot tear or glue it. It need not respect distances, derivatives, or the complex coordinates.

Family 059 claims a negative answer in two forms. One construction produces multiplicities two and three in a finite higher dimension. Another produces multiplicities four and five in exactly four complex variables. The distinction from a continuously varying family is essential: strong positive equimultiplicity theorems for families remain compatible with these claims about isolated pairs.

2. Background and essential definitions

A germ and its order

A function germ at the origin records a holomorphic function only on a sufficiently small neighborhood; shrinking the neighborhood does not change the germ. Write its convergent power series as

$$f(z)=f_m(z)+f_{m+1}(z)+\cdots,$$

where each $f_j$ is homogeneous of ordinary total degree $j$ and $f_m\ne0$. Its order is $\operatorname{ord}_0f=m$. If $f$ is reduced, meaning it has no repeated irreducible factor, this is the multiplicity of its zero-set germ $V(f)$.

For example, $x^2+y^3$ has multiplicity two. Assigning different weights to variables can make it weighted homogeneous, but those weights do not change its ordinary multiplicity. The manuscript’s weighted constructions always retain ordinary total degree for this question.

Two hypersurface germs are ambiently homeomorphic if a homeomorphism near the origin in $\mathbb C^N$ carries one zero set to the other. A stronger statement about functions is topological right equivalence:

$$f=g\circ\phi$$

for an ambient homeomorphism germ $\phi$. This preserves all nearby level sets, not just the zero level. Neither notion requires $\phi$ to be holomorphic.

The topology of an isolated singularity

An isolated critical point means the gradient vanishes only at the origin in a sufficiently small neighborhood. Intersecting the zero set with a small sphere gives its link. Nearby nonzero fibers of the function give the Milnor fiber, whose essential topology is a bouquet of spheres in the middle dimension.

The number of those spheres is the Milnor number,

$$\mu(f)=\dim_{\mathbb C} \frac{\mathbb C\{z_1,\ldots,z_N\}} {(\partial f/\partial z_1,\ldots,\partial f/\partial z_N)}.$$

Moving the value of $f$ once around zero gives a monodromy transformation on the fiber’s homology. Its Seifert form records linking of a cycle with another cycle pushed into a neighboring fiber. This is an integer-valued bilinear form on a lattice, so preserving it requires more than matching dimensions or eigenvalues.

3. Development history and previous partial results

Zariski’s work on plane curves showed how embedded topology determines branch data and multiplicity; he posed the higher-dimensional question explicitly in 1971. The A’Campo–Lê theorem also shows that an ambiently topologically smooth reduced hypersurface is analytically smooth. Thus multiplicity one cannot participate in an unequal-multiplicity counterexample. The manuscripts cite these original sources.

The deformation problem developed alongside the pair problem. Lê–Ramanujam proved that a one-parameter holomorphic family of isolated hypersurface singularities with constant Milnor number is topologically constant outside the exceptional ambient dimension three. Greuel and O’Shea proved equimultiplicity in the quasihomogeneous family setting. Fernández de Bobadilla–Pełka later established the general equimultiplicity theorem for continuous families with finite constant Milnor number, including coefficientwise continuous formal families. Fernández de Bobadilla–Pełka.

That theorem does not identify all pairs with the same topology as members of one such family. Their work already gives topologically equivalent four-variable germs of equal multiplicity that cannot be connected by a constant-Milnor-number family. This provides a concrete precedent for separating the two questions.

Additional geometric structure restores positive results in other settings. McLean recovered multiplicity from embedded contact information on the link, which remembers a natural field of tangent hyperplanes beyond the ordinary embedded topology. Ambient bi-Lipschitz equivalence, preserving distances up to fixed factors, also preserves multiplicity two in the result cited by M1. These are stronger equivalence relations than the homeomorphisms claimed here. McLean.

The Seifert-form route has its own history. Du Bois–Michel constructed plane curves with isometric integral Seifert forms but different embedded types. In higher dimensions, knot and open-book classification theorems allow such forms to determine more topology. Both manuscripts exploit this dimension-sensitive passage, with different constructions and classification inputs.

4. What the new manuscripts claim

M1: multiplicities two and three

M1 claims that there is a finite integer $N>3$, divisible by eight, and reduced real-coefficient polynomials $f_1,f_2$ in $N$ complex variables such that both are weighted homogeneous with positive rational weights, both have isolated critical points, and

$$\operatorname{ord}_0f_1=2,\qquad \operatorname{ord}_0f_2=3,$$

while

$$(\mathbb C^N,V(f_1),0)\cong (\mathbb C^N,V(f_2),0)$$

as topological ambient-pair germs. The construction determines a finite dimension through integer arithmetic; the theorem does not minimize it.

The same pair is claimed to be topologically right equivalent. The homeomorphism for right equivalence may differ from the first ambient-pair map. Since two and three have different parity, this also refutes the proposed parity-only multiplicity invariant for right equivalence.

The Hessian corank is $N-\operatorname{rank}_{\mathbb C}\operatorname{Hess}_0(f)$, the number of directions not detected by the quadratic form. M1 claims unequal coranks for this pair, giving a negative answer to the ambient-topological version of Arnold’s corank question. Their coranks have the same parity, consistent with the separate parity invariant cited by the source.

One more consequence uses the initial homogeneous part $f_*$. After adding one common square in a new variable, the two functions remain right equivalent, but the unit fibers of their initial forms have different numbers of connected components: one is connected and the other has two components. Their initial-form Milnor fibers therefore cannot be homotopy equivalent. The full initial polynomial is used even when it is nonreduced.

M2: multiplicities four and five in four variables

M2 claims reduced convergent holomorphic germs $f_4,f_5$ on $(\mathbb C^4,0)$, each with an isolated critical point, satisfying

$$\operatorname{ord}_0f_4=4,\qquad \operatorname{ord}_0f_5=5,$$

whose zero sets are ambiently homeomorphic.

This is a distinct construction. Its equations are convergent germs obtained in analytic charts, rather than the real-coefficient weighted homogeneous polynomials of M1. Its stated main conclusion is ambient-pair equivalence; the additional right-equivalence and corank consequences above belong to M1 and are not silently transferred to M2.

The hypersurfaces in $\mathbb C^4$ have complex dimension three. Neither manuscript treats the ambient three-variable case, whose hypersurfaces are complex surfaces.

5. Why the advance matters

Importance: 5/5. If correct, these would answer one of the central questions linking equations and topology of singularities. Multiplicity is among the first algebraic invariants one computes, so showing that even embedded topology can forget it would substantially change the classification picture.

The four-variable construction is especially significant because it brings the proposed failure close to the remaining low-dimensional cases. The higher-dimensional construction supplies the stronger function-level equivalence and the smallest possible unequal multiplicity pair involving a singularity of multiplicity two.

The consequences also clarify what extra information matters. Contact geometry, metric control, and a continuous constant-Milnor-number path are not decorative enhancements of topology; they can preserve information lost by an arbitrary ambient homeomorphism.

6. The main idea

Keep the lowest degree while matching an integral form

The common goal is to arrange the same integral Seifert form without equalizing ordinary multiplicity. Matching real forms or monodromy eigenvalues alone is insufficient: an integral lattice can retain arithmetic differences invisible over the real numbers.

M1 starts from finite multiplicative relations among rational numbers and turns them into weighted chain polynomials. It balances their different numbers of variables using squares on one side, while common cubes keep the other side at multiplicity three. These constructions preserve the desired difference in lowest degree while arranging preliminary spectral and monodromy comparisons.

Adding polynomials in disjoint variables corresponds, through Thom–Sebastiani and Sakamoto’s formulas, to tensoring the relevant homology lattices and Seifert forms. The manuscript uses carefully chosen common powers and tensor operations to improve compatibility. It obtains isometries over the rationals and over the integers localized at every prime, then uses strong approximation for appropriate algebraic groups to assemble one integral isometry.

Strong approximation is an arithmetic principle allowing a rational group element to meet several local conditions simultaneously. Here its role is to make all the local lattice comparisons agree with a single global integral map. The construction’s substantial arithmetic work lies between a suggestive spectral coincidence and the topological conclusion.

Finally, the source arranges spherical links in a range where Levine’s classification applies. The integral Seifert-form comparison yields embedded link equivalence, and weighted radial coordinates extend it to the ambient germs. Further results of King and Saeki supply the right-equivalence statement.

A four-variable construction with mapped plane curves

M2 begins with the plane curve

$$h(X,T)=T(X-T^2)(X-2T^2)$$

and two degree-four finite maps to its plane. The pulled-back curves have the same six-branch embedded type. The important difference is in their maps to the common base: the induced homology pushforwards carry information beyond the abstract isometry of the plane Seifert forms.

The source inserts these data into smoothings related to the cone $yz=w^4$. Analytic elimination produces the two required orders. To compute the Milnor page, it separates an inner region from an exterior region and matches their cycles along the boundary. The matching condition depends on residues modulo four and on the particular finite map.

Two resulting lattice splittings give rational comparisons that are integral at complementary primes. Another strong-approximation argument combines them into an integral Seifert-form congruence. Kato’s classification of the relevant open books on $S^7$ then supplies embedded link equivalence without requiring spherical bindings. An analytic conical-structure theorem extends this to the germs.

These are the manuscripts’ proposed constructions and uses of classification theorems, not independent checks of their lattice arithmetic.

7. Scope, limitations, and open questions

Both assertions concern isolated hypersurface singularities and ordinary ambient topology. Neither produces the claimed unequal multiplicities under a bi-Lipschitz or contact-preserving equivalence. M1 explicitly does not claim its homeomorphism has that extra regularity.

The pairs cannot be connected by the constant-Milnor-number continuous families covered by Fernández de Bobadilla–Pełka, because their multiplicities differ. Thus the family theorem remains a positive result and cannot be used to infer a contradiction merely from equal topology of two endpoints.

The ambient three-variable problem is untouched. M1 does not identify the smallest possible dimension for its two-versus-three construction, and M2 does not claim a two-versus-three pair in four variables. Determining how low the dimension and multiplicities can be, and identifying additional structures sufficient to recover multiplicity, remain natural questions.

The initial-form consequence concerns connectedness of the initial-form fibers. Isolatedness is asserted for the stabilized original germs, not automatically for their lowest homogeneous parts.

8. Sources and further reading

Both manuscripts state their main counterexamples in Theorem 1.1. M1’s Corollaries 6.3–6.5 cover corank, right equivalence, and initial-form fibers. Its Sections 3–6 separate the seed construction, arithmetic compatibility, and topological transfer. M2’s Sections 2–9 develop the finite-map data and integral gluing before applying open-book classification.

For the positive family result, see Fernández de Bobadilla–Pełka. Sampaio on initial-form Milnor fibers explains the related conjectures. The manuscripts’ bibliographies give Zariski’s original question, Milnor’s foundational fibration theorem, and the knot-classification inputs.

Source manuscripts

• M1. OpenAI, Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three (September 24, 2026).
• M2. OpenAI, Ambiently homeomorphic isolated hypersurface germs in ℂ⁴ with multiplicities four and five (September 27, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.