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.
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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
initial ideals ⓘ secondary fan ⓘ state polytope ⓘ term orders ⓘ tropical variety ⓘ weight vectors ⓘ |
| relatesTermOrdersVia | adjacency of Gröbner cones GENERATED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.