Ring of polynomial functions
In mathematics, the ring of polynomial functions on a vector space gives a coordinate-free analog of a polynomial ring. It can be motivated as follows. If , then, as the notation suggests, we can view as coordinate functions on : when This suggests the following: let V be a finite-dimensional vector space over an infinite field k and then let S be the subring generated by the dual space of the ring of functions . If we fix a basis for V and write for its dual basis, then S consists of polynomials in ; S can be viewed as a polynomial ring over k. If V is viewed as an algebraic variety, then this S is precisely the coordinate ring of V.
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