Malgrange preparation theorem
E1020443
The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Malgrange preparation theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T13070901 — 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: Malgrange preparation theorem Context triple: [Bernard Malgrange, notableFor, Malgrange preparation theorem]
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A.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
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B.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
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C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
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D.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
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E.
Puiseux series
Puiseux series are formal power series in fractional powers of a variable, widely used in algebraic geometry and singularity theory to locally parametrize algebraic curves.
- 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: Malgrange preparation theorem Target entity description: The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
-
A.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
B.
Mittag-Leffler theorem
The Mittag-Leffler theorem is a fundamental result in complex analysis that characterizes meromorphic functions by allowing the construction of such functions with prescribed principal parts at given poles.
-
C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Bochner–Martinelli formula
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
-
E.
Puiseux series
Puiseux series are formal power series in fractional powers of a variable, widely used in algebraic geometry and singularity theory to locally parametrize algebraic curves.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in analysis ⓘ theorem in singularity theory ⓘ |
| appliesTo |
C-infinity functions
ⓘ
germs of smooth functions ⓘ real-analytic functions ⓘ smooth functions ⓘ |
| assumes | smoothness conditions on coefficients ⓘ |
| concerns | local behavior of smooth mappings ⓘ |
| context |
ideals generated by smooth functions
ⓘ
local rings of smooth function germs ⓘ |
| describes | structure of smooth functions near singular points ⓘ |
| field |
differential topology
ⓘ
mathematical analysis ⓘ microlocal analysis ⓘ singularity theory ⓘ |
| generalizes | Weierstrass preparation theorem NERFINISHED ⓘ |
| hasConsequence |
finite generation of certain modules of smooth functions
ⓘ
local polynomial representation in a distinguished variable ⓘ |
| hasDomain | germs of smooth functions on Euclidean spaces ⓘ |
| holdsIn |
complex smooth category
ⓘ
real smooth category ⓘ |
| implies |
existence of invertible smooth factors
ⓘ
existence of polynomial-like factors ⓘ |
| isPartOf |
local analysis of mappings
ⓘ
theory of singularities of differentiable maps ⓘ |
| namedAfter | Bernard Malgrange NERFINISHED ⓘ |
| provides | local factorization of smooth functions ⓘ |
| relatedTo |
Malgrange division theorem
NERFINISHED
ⓘ
Thom–Mather theory NERFINISHED ⓘ Weierstrass division theorem NERFINISHED ⓘ division theorem for smooth functions ⓘ finite determinacy of singularities ⓘ |
| strengthens | Weierstrass preparation theorem for analytic functions NERFINISHED ⓘ |
| usedFor |
analyzing multiplicity of zeros of smooth functions
ⓘ
constructing local models of singularities ⓘ local study of solutions of PDEs near characteristic points ⓘ proving stability results in singularity theory ⓘ reducing smooth maps to polynomial form in one variable ⓘ |
| usedIn |
implicit function problems
ⓘ
local normal form theory ⓘ microlocal analysis of PDEs ⓘ singularity classification ⓘ study of differential equations ⓘ theory of stratifications ⓘ transversality arguments ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Malgrange preparation theorem Description of subject: The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.