Bernstein–Sato polynomial
E934438
The Bernstein–Sato polynomial is a fundamental object in algebraic analysis and singularity theory that encodes deep information about the behavior of functions and their singularities via differential equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bernstein–Sato polynomial canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11576265 — 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: Bernstein–Sato polynomial Context triple: [Joseph Bernstein, notableWork, Bernstein–Sato polynomial]
-
A.
Hilbert polynomial
The Hilbert polynomial is an algebraic invariant that encodes the asymptotic growth of the dimension of graded components of a module or the number of independent conditions imposed by a projective variety.
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B.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
-
C.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
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D.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
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E.
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.
- 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: Bernstein–Sato polynomial Target entity description: The Bernstein–Sato polynomial is a fundamental object in algebraic analysis and singularity theory that encodes deep information about the behavior of functions and their singularities via differential equations.
-
A.
Hilbert polynomial
The Hilbert polynomial is an algebraic invariant that encodes the asymptotic growth of the dimension of graded components of a module or the number of independent conditions imposed by a projective variety.
-
B.
Symanzik polynomials
Symanzik polynomials are graph-based polynomials that arise in the parametric representation of Feynman integrals in quantum field theory, encoding the topology and kinematic dependence of Feynman diagrams.
-
C.
Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
-
D.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
invariant of singularities
ⓘ
mathematical object ⓘ polynomial ⓘ |
| alsoKnownAs |
Bernstein polynomial of a function
ⓘ
b-function ⓘ |
| appearsIn |
Hodge theory of singularities
ⓘ
theory of perverse sheaves ⓘ
linked to:
theory of D-modules
|
| associatedWith |
each nonzero holomorphic function germ
ⓘ
each nonzero polynomial function ⓘ |
| centralConceptIn |
algebraic analysis of differential equations
ⓘ
study of singularities via D-modules ⓘ |
| definedOver |
complex analytic functions
ⓘ
polynomials over C ⓘ |
| dependsOn |
holomorphic function
ⓘ
polynomial function ⓘ |
| encodes |
behavior of a function under differential operators
ⓘ
information about singularities of a function ⓘ monodromy information of singularities ⓘ spectrum of singularities ⓘ |
| field |
D-module theory
ⓘ
algebraic analysis ⓘ algebraic geometry ⓘ microlocal analysis ⓘ singularity theory ⓘ |
| generalizationOf | Bernstein polynomial for one-variable polynomials ⓘ |
| historicalDevelopment | introduced in the 1970s ⓘ |
| namedAfter |
Ilya N. Bernstein
ⓘ
Mikio Sato NERFINISHED ⓘ |
| property |
degree depends on complexity of singularity
ⓘ
has rational roots ⓘ roots are negative rational numbers for many classes of functions ⓘ |
| relatedTo |
D-modules
ⓘ
linked to:
theory of D-modules
Igusa zeta function ⓘ Malgrange’s formula ⓘ V-filtration ⓘ holonomic D-modules ⓘ local zeta function ⓘ log canonical threshold ⓘ microlocal b-function ⓘ multiplier ideals ⓘ |
| satisfies | functional equation with differential operator P(s) ⓘ |
| usedFor |
analyzing asymptotic behavior of integrals
ⓘ
computing invariants of singular points ⓘ studying monodromy of Milnor fibers ⓘ studying singularities of hypersurfaces ⓘ |
| variable | s ⓘ |
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: Bernstein–Sato polynomial Description of subject: The Bernstein–Sato polynomial is a fundamental object in algebraic analysis and singularity theory that encodes deep information about the behavior of functions and their singularities via differential equations.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.