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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.
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.
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