Hilbert polynomial
E790523
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hilbert polynomial canonical | 2 |
| Hilbert–Samuel polynomial | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9297101 — 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: Hilbert polynomial Context triple: [Castelnuovo–Mumford regularity, relatedTo, Hilbert polynomial]
-
A.
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.
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B.
Milnor number
The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
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C.
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.
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D.
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.
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E.
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.
- 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: Hilbert polynomial Target entity description: 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.
-
A.
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.
-
B.
Milnor number
The Milnor number is an invariant in singularity theory that measures the complexity of an isolated critical point of a complex hypersurface or function.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
polynomial ⓘ |
| agreesWith | Hilbert function for sufficiently large degree ⓘ |
| appearsIn |
Hilbert’s basis theorem context
ⓘ
Hilbert’s work on syzygies ⓘ |
| associatedWith |
Noetherian graded ring
ⓘ
graded module ⓘ homogeneous ideal ⓘ projective variety ⓘ |
| canBeComputedBy | Gröbner basis methods ⓘ |
| computedFrom |
Betti numbers
ⓘ
minimal graded free resolution ⓘ |
| definedFor | finitely generated graded module ⓘ |
| definedOver | graded ring ⓘ |
| dependsOn | choice of embedding into projective space ⓘ |
| describes |
asymptotic growth of Hilbert function
ⓘ
asymptotic growth of dimensions of graded components of a module ⓘ |
| encodes |
dimension of graded components for large degree
ⓘ
number of independent conditions imposed by a projective variety ⓘ |
| field |
algebraic geometry
ⓘ
commutative algebra ⓘ |
| generalizes | dimension counting in linear systems ⓘ |
| hasApplication |
bounding number of generators of ideals
ⓘ
moduli problems in algebraic geometry ⓘ |
| hasCoefficient | leading coefficient related to degree of variety ⓘ |
| hasInput | nonnegative integer degree ⓘ |
| hasLeadingTerm | (degree of variety)/(dimension!)·n^{dimension} ⓘ |
| hasOutput | dimension of graded component for large degree ⓘ |
| hasProperty |
degree equals dimension of projective variety
ⓘ
takes integer values for integer arguments ⓘ |
| hasSpecialCase | Hilbert–Samuel polynomial NERFINISHED ⓘ |
| introducedBy | David Hilbert NERFINISHED ⓘ |
| invariantOf | projective scheme up to isomorphism ⓘ |
| invariantUnder | projective isomorphism ⓘ |
| relatedTo |
Hilbert series
NERFINISHED
ⓘ
Poincaré series NERFINISHED ⓘ |
| relatesTo | Hilbert function NERFINISHED ⓘ |
| usedIn |
Castelnuovo–Mumford regularity theory
NERFINISHED
ⓘ
classification of projective varieties ⓘ computational algebraic geometry ⓘ construction of Hilbert scheme ⓘ intersection theory ⓘ |
| usedToDefine |
arithmetic genus
ⓘ
degree of projective variety ⓘ geometric invariants of projective schemes ⓘ |
| usedToStudy |
families of projective varieties
ⓘ
flat families of schemes ⓘ |
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: Hilbert polynomial Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Introduction to Commutative Algebra
this entity surface form:
Hilbert–Samuel polynomial