Weierstrass preparation theorem

E112259

mathematical theorem theorem in analytic geometry theorem in complex analysis

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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All labels observed (8)

Label	Occurrences
Weierstrass preparation theorem canonical	3
Weierstrass division theorem	2
Malgrange division theorem	1
Weierstrass polynomial	1
Weierstrass preparation over complete local rings	1
Weierstrass preparation theorem for analytic functions	1
formal Weierstrass preparation theorem	1
non-Archimedean Weierstrass preparation theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T940261 — 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: Weierstrass preparation theorem
Context triple: [Karl Weierstrass, notableFor, Weierstrass preparation theorem]

A. Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
B. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
C. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
D. Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
E. Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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: Weierstrass preparation theorem
Target entity description: 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.

A. Weierstrass factorization theorem
The Weierstrass factorization theorem is a fundamental result in complex analysis that expresses any entire function as an infinite product determined by its zeros, generalizing the factorization of polynomials.
B. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
C. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
D. Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
E. Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in analytic geometry ⓘ theorem in complex analysis ⓘ
appliesTo	analytic functions ⓘ convergent power series ⓘ holomorphic functions ⓘ
assumes	analyticity in a neighborhood of the point ⓘ non-flatness in one distinguished variable ⓘ
category	result in local analytic geometry ⓘ
coreIdea	expression of analytic function as product of polynomial and unit ⓘ local factorization of analytic functions near a zero ⓘ power-series analogue of polynomial factorization ⓘ
ensures	finite multiplicity of zeros in the distinguished variable ⓘ local finite mapping property over the distinguished coordinate ⓘ
field	analytic geometry ⓘ commutative algebra ⓘ complex analysis ⓘ local analytic geometry ⓘ singularity theory ⓘ
generalizationOf	factorization of polynomials over fields ⓘ
hasVariant	Weierstrass division theorem ⓘ linked to: Weierstrass preparation theorem Weierstrass preparation over complete local rings ⓘ linked to: Weierstrass preparation theorem formal Weierstrass preparation theorem ⓘ linked to: Weierstrass preparation theorem non-Archimedean Weierstrass preparation theorem ⓘ linked to: Weierstrass preparation theorem
holdsIn	ring of convergent power series in several complex variables ⓘ
implies	analytic sets are locally finite over a coordinate ⓘ local finite generation of certain analytic modules ⓘ structure theorem for zeros of analytic functions ⓘ
involvesConcept	Weierstrass polynomial ⓘ linked to: Weierstrass preparation theorem analytic local algebra ⓘ distinguished polynomial ⓘ local ring of convergent power series ⓘ order of vanishing ⓘ regularity in one variable ⓘ unit in a local ring ⓘ
namedAfter	Karl Weierstrass ⓘ
relatedTo	Cartan theorems A and B ⓘ Noether normalization lemma ⓘ Oka coherence theorem ⓘ implicit function theorem ⓘ
statesThat	a suitably regular analytic function near a point can be written as a Weierstrass polynomial times a unit ⓘ
typicalForm	f(z,w)=u(z,w)P(z,w) with u a unit and P a monic polynomial in one variable with analytic coefficients ⓘ
usedFor	establishing properties of analytic algebras ⓘ local factorization of holomorphic mappings ⓘ parametrizing branches of analytic curves ⓘ proving Weierstrass division theorem ⓘ resolving singularities locally in analytic geometry ⓘ studying local structure of analytic sets ⓘ

How these facts were elicited