Hilbert polynomial

E790523

algebraic invariant 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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Observed surface forms (1)

Surface form	Occurrences
Hilbert–Samuel polynomial	1

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 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Castelnuovo–Mumford regularity → relatedTo → Hilbert polynomial ⓘ

“Introduction to Commutative Algebra” (with Ian G. Macdonald) → hasSubject → Hilbert polynomial ⓘ

subject surface form: Introduction to Commutative Algebra

this entity surface form: Hilbert–Samuel polynomial

Gelfand–Kirillov dimension → relatedTo → Hilbert polynomial ⓘ