Milnor number
E265517
The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Milnor number canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T2418316 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Milnor number Context triple: [John Milnor, notableWork, Milnor number]
-
A.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
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C.
Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
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D.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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E.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Milnor number Target entity description: The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
-
A.
Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
-
B.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
-
C.
Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
-
D.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
E.
Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical invariant
ⓘ
singularity invariant ⓘ |
| appearsIn |
Gauss–Bonnet type formulas for singular hypersurfaces
ⓘ
local Euler characteristic formulas ⓘ |
| appliesTo |
complex hypersurface singularity
ⓘ
holomorphic function germ ⓘ isolated critical point ⓘ isolated hypersurface singularity ⓘ |
| characterizes | local behavior near an isolated critical point ⓘ |
| condition | finite if and only if the critical point is isolated ⓘ |
| definition | dimension over ℂ of the local algebra ℂ{x₁,…,x_n}/J_f ⓘ |
| equals |
number of n-dimensional vanishing cycles in the Milnor fiber
ⓘ
rank of the middle homology of the Milnor fiber ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ differential topology ⓘ singularity theory ⓘ |
| forFunction | f(z)=z^{k+1} has μ = k ⓘ |
| forMorseSingularity | μ = 1 ⓘ |
| forNondegenerateConvenientPlaneCurve | computable from Newton polygon ⓘ |
| forNonSingularPoint | μ = 0 ⓘ |
| forPlaneCurveSingularities | μ = 2δ − r + 1 where δ is delta invariant and r is number of branches ⓘ |
| generalization |
Bruce–Roberts number
ⓘ
Lê numbers ⓘ Milnor number of complete intersection singularities ⓘ |
| inequality | Tjurina number ≤ Milnor number ⓘ |
| introducedBy | John Milnor ⓘ |
| introducedIn | 1960s ⓘ |
| isDefinedFor | germ of a holomorphic function f:(ℂ^n,0)→(ℂ,0) ⓘ |
| measures | complexity of a singularity ⓘ |
| namedAfter | John Milnor ⓘ |
| property |
invariant under analytic equivalence of function germs
ⓘ
topological invariant of the germ of an isolated hypersurface singularity ⓘ |
| relatedInvariant | Tjurina number ⓘ |
| relatedTo |
Jacobian algebra
ⓘ
Milnor fibration ⓘ intersection multiplicity of polar curves ⓘ monodromy of a singularity ⓘ vanishing cycles ⓘ |
| symbol | μ ⓘ |
| usedIn |
ADE singularity theory
ⓘ
classification of simple singularities ⓘ deformation theory of singularities ⓘ equisingularity theory ⓘ singularity theory of complex analytic maps ⓘ |
| uses | Jacobian ideal J_f generated by partial derivatives of f ⓘ |
How these facts were elicited
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Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Milnor number Description of subject: The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.