Puiseux series
E627726
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Puiseux series canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6910708 — 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: Puiseux series Context triple: [Hahn series, generalizes, Puiseux series]
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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.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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C.
Hadamard product (of power series)
The Hadamard product (of power series) is an operation that forms a new power series by multiplying the corresponding coefficients of two given power series term by term.
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D.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
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E.
Essai sur l’étude des fonctions données par leur développement de Taylor
Essai sur l’étude des fonctions données par leur développement de Taylor is a foundational mathematical treatise by Jacques Hadamard that investigates the behavior and properties of functions defined through their Taylor series expansions.
- 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: Puiseux series Target entity description: 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.
-
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.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
-
C.
Hadamard product (of power series)
The Hadamard product (of power series) is an operation that forms a new power series by multiplying the corresponding coefficients of two given power series term by term.
-
D.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
-
E.
Essai sur l’étude des fonctions données par leur développement de Taylor
Essai sur l’étude des fonctions données par leur développement de Taylor is a foundational mathematical treatise by Jacques Hadamard that investigates the behavior and properties of functions defined through their Taylor series expansions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
formal power series
ⓘ
mathematical concept ⓘ |
| appearsIn |
computation of discriminants of polynomials
ⓘ
local study of plane curve singularities ⓘ study of analytic continuation of algebraic functions ⓘ |
| componentOf | local field of an algebraic curve at a branch ⓘ |
| constraint |
denominators of exponents are bounded
ⓘ
set of exponents is well-ordered ⓘ |
| definedOver | algebraically closed field of characteristic zero ⓘ |
| fieldOfStudy |
algebraic geometry
ⓘ
commutative algebra ⓘ singularity theory ⓘ valuation theory ⓘ |
| generalizationOf | formal power series in integral powers ⓘ |
| hasForm | sum a_i x^{q_i} with rational exponents q_i ⓘ |
| hasPart |
coefficients from a base field or ring
ⓘ
fractional powers of a variable ⓘ |
| hasVariant |
generalized Puiseux series
ⓘ
multivariate Puiseux series ⓘ |
| historicalDevelopment | introduced in the 19th century ⓘ |
| namedAfter | Victor Puiseux ⓘ |
| property |
allows rational exponents with bounded denominators
ⓘ
can be truncated to obtain finite Puiseux polynomials ⓘ forms an algebraically closed field over an algebraically closed base field of characteristic zero ⓘ supports Cauchy product multiplication ⓘ supports termwise addition ⓘ totally ordered by exponent of the first nonzero term ⓘ |
| relatedTo |
Abhyankar valuations
ⓘ
Henselian local fields ⓘ
linked to:
Henselian ring
Laurent series ⓘ Newton polygon ⓘ Weierstrass preparation theorem ⓘ ramification of coverings ⓘ resolution of singularities of plane curves ⓘ tropical geometry via valuations ⓘ valuation on function fields ⓘ |
| usedFor |
Newton–Puiseux algorithm
ⓘ
computing intersection multiplicities ⓘ computing semigroup of values of a plane branch ⓘ describing branches of plane algebraic curves ⓘ describing monodromy of algebraic functions ⓘ expressing algebraic functions as series expansions ⓘ local parametrization of algebraic curves ⓘ resolution of curve singularities ⓘ |
| usedIn |
algorithmic algebraic geometry
ⓘ
computation of branches of polynomial equations in two variables ⓘ computer algebra systems for curve analysis ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: Puiseux series Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.