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.

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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)
sameAs	Ring of polynomial functions Ring of polynomial functions
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Clarify dbt:Main dbt:Reflist dbt:Hair_space dbt:Math_theorem
Subject	Polynomial functions Ring theory
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Ring_of_polynomial_functions?oldid=932232125&ns=0
Wikipage page ID	38608533 (xsd:integer)
page length (characters) of wiki page	9001 (xsd:nonNegativeInteger)
Wikipage revision ID	932232125 (xsd:integer)
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)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Ring_of_polynomial_functions
is rdfs:seeAlso of	Polynomial
is Wikipage redirect of	Polynomials on vector spaces
is Link from a Wikipage to another Wikipage of	Geometric invariant theory Glossary of algebraic geometry Chern–Weil homomorphism First and second fundamental theorems of invariant theory Fixed-point subgroup David Hilbert

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