About: Ring of polynomial functions

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In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. If k is infinite, then k[V] is the symmetric algebra of the dual space . In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies.

Attributes	Values
rdf:type	yago:Abstraction100002137 yago:Function113783816 yago:MathematicalRelation113783581 yago:Polynomial105861855 yago:Relation100031921 yago:WikicatPolynomials
rdfs:label	Ring of polynomial functions (en)
rdfs:comment	In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. If k is infinite, then k[V] is the symmetric algebra of the dual space . In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies. (en)
dct:subject	Ring theory Polynomial functions
Wikipage page ID	38608533 (xsd:integer)
Wikipage revision ID	1117893810 (xsd:integer)
Link from a Wikipage to another Wikipage	Basis (linear algebra) Algebra homomorphism Algebraic geometry of projective spaces Vector space Degree of a polynomial Ring theory Complex number Mathematics General linear group Operator product expansion Subring Endomorphism Function (mathematics) Polynomial functions Smooth function Bijective Zariski tangent space Partial derivative Physics Linear map Algebraic variety Dual space Field (mathematics) Dimension (vector space) Foundations of Differential Geometry Isomorphism Quadratic form Ring (mathematics) Taylor series Surjective Characteristic (algebra) Symmetric algebra Homogeneous polynomial Associative algebra Polynomial Polynomial ring Field extension Coordinate ring Radius of convergence Real number Set (mathematics) Section (fiber bundle) Vector bundle Injective Ring of functions Lagrange interpolation Structure constant
sameAs	Ring of polynomial functions Ring of polynomial functions Ring of polynomial functions Ring of polynomial functions
dbp:wikiPageUsesTemplate	dbt:Hair_space dbt:Citation dbt:Clarify dbt:Main dbt:Reflist dbt:Math_theorem
has abstract	In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] be the commutative k-algebra generated by the dual space , which is a subring of the ring of all functions . If we fix a basis for V and write for its dual basis, then k[V] consists of polynomials in . If k is infinite, then k[V] is the symmetric algebra of the dual space . In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies. Throughout the article, for simplicity, the base field k is assumed to be infinite. (en)
prov:wasDerivedFrom	wikipedia-en:Ring_of_polynomial_functions?oldid=1117893810&ns=0
page length (characters) of wiki page	9038 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Ring_of_polynomial_functions
is rdfs:seeAlso of	Polynomial
is Link from a Wikipage to another Wikipage of	Projectivization Chern–Weil homomorphism Operator product expansion Geometric invariant theory Glossary of algebraic geometry Invariant theory Tensor product of representations Polynomial ring First and second fundamental theorems of invariant theory Singular point of a curve Petkovšek's algorithm Polynomials on vector spaces
is Wikipage redirect of	Polynomials on vector spaces
is foaf:primaryTopic of	wikipedia-en:Ring_of_polynomial_functions

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