Gröbner fan
E838598
A Gröbner fan is a polyhedral fan that encodes all initial ideals (and thus all Gröbner bases) of an ideal with respect to different term orders.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gröbner cone | 1 |
| Gröbner fan canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10055765 — 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: Gröbner fan Context triple: [Gröbner basis, relatedConcept, Gröbner fan]
-
A.
Gröbner basis
A Gröbner basis is a particular generating set of an ideal in a polynomial ring that allows algorithmic solutions to many problems in computational algebra, such as ideal membership and solving systems of polynomial equations.
-
B.
Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
-
C.
Knuth–Bendix order
The Knuth–Bendix order is a well-founded, total, simplification ordering on terms used in automated theorem proving and term rewriting systems to ensure termination and confluence.
-
D.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
E.
Leegebruch
Leegebruch is a small municipality in the German state of Brandenburg, located in the Oberhavel district just north of Berlin.
- 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: Gröbner fan Target entity description: A Gröbner fan is a polyhedral fan that encodes all initial ideals (and thus all Gröbner bases) of an ideal with respect to different term orders.
-
A.
Gröbner basis
A Gröbner basis is a particular generating set of an ideal in a polynomial ring that allows algorithmic solutions to many problems in computational algebra, such as ideal membership and solving systems of polynomial equations.
-
B.
Buchberger algorithm
The Buchberger algorithm is a fundamental procedure in computational algebra for computing Gröbner bases of polynomial ideals, enabling systematic solutions to systems of polynomial equations.
-
C.
Knuth–Bendix order
The Knuth–Bendix order is a well-founded, total, simplification ordering on terms used in automated theorem proving and term rewriting systems to ensure termination and confluence.
-
D.
Knuth–Bendix completion algorithm
The Knuth–Bendix completion algorithm is a procedure in term rewriting and automated theorem proving that transforms a set of equations into a confluent rewriting system, enabling decision of word problems in algebraic structures.
-
E.
Leegebruch
Leegebruch is a small municipality in the German state of Brandenburg, located in the Oberhavel district just north of Berlin.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
combinatorial object
ⓘ
concept in computational algebraic geometry ⓘ geometric object ⓘ polyhedral fan ⓘ |
| ambientSpace | real vector space of weight vectors ⓘ |
| arisesFrom | initial ideal construction with respect to weight vectors ⓘ |
| associatedTo | toric degenerations of varieties ⓘ |
| captures | combinatorial structure of all term orders for an ideal ⓘ |
| computableBy |
Gröbner walk algorithm
ⓘ
reverse search over cones ⓘ |
| coneRepresents | set of weight vectors with same initial ideal ⓘ |
| definedFor | ideal in a polynomial ring ⓘ |
| dependsOn | given ideal ⓘ |
| dimensionOfConesDependsOn | number of variables in the polynomial ring ⓘ |
| eachGröbnerConeCorrespondsTo | one reduced Gröbner basis ⓘ |
| encodes |
all Gröbner bases of an ideal
ⓘ
all initial ideals of an ideal ⓘ |
| generalizes | Gröbner region of an ideal ⓘ |
| hasApplicationIn |
algorithmic algebraic geometry software
ⓘ
symbolic computation ⓘ |
| hasCone |
Gröbner cone
ⓘ
linked to:
Gröbner fan
|
| hasFacePosetRelatedTo | poset of initial ideals ⓘ |
| introducedInField | computational commutative algebra ⓘ |
| invariantUnder | change of coordinates preserving the ideal ⓘ |
| isCompleteFanIn | space of all weight vectors modulo scaling ⓘ |
| isRefinementOf | normal fan of the state polytope ⓘ |
| maximalConesCorrespondTo | distinct reduced Gröbner bases ⓘ |
| property |
complete fan
ⓘ
cones correspond to distinct initial ideals ⓘ polyhedral fan ⓘ |
| relatedTo |
Gröbner bases
ⓘ
linked to:
Gröbner basis
initial ideals ⓘ secondary fan ⓘ state polytope ⓘ term orders ⓘ tropical variety ⓘ weight vectors ⓘ |
| relatesTermOrdersVia | adjacency of Gröbner cones ⓘ |
| studiedIn |
polyhedral geometry
ⓘ
theory of Gröbner bases ⓘ |
| usedIn |
algebraic geometry
ⓘ
classification of Gröbner bases ⓘ combinatorial commutative algebra ⓘ computing all reduced Gröbner bases of an ideal ⓘ study of term orders ⓘ tropical geometry ⓘ |
| usedToStudy |
deformations of ideals
ⓘ
flat degenerations to monomial ideals ⓘ |
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: Gröbner fan Description of subject: A Gröbner fan is a polyhedral fan that encodes all initial ideals (and thus all Gröbner bases) of an ideal with respect to different term orders.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gröbner fan