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.

The explicit definition of the ring can be given as follows. If ${\displaystyle k[t_{1},\dots ,t_{n}]}$ is a polynomial ring, then we can view ${\displaystyle t_{i}}$ as coordinate functions on ${\displaystyle k^{n}}$; i.e., ${\displaystyle t_{i}(x)=x_{i}}$ when ${\displaystyle x=(x_{1},\dots ,x_{n}).}$ This suggests the following: given a vector space V, let k[V] be the commutative k-algebra generated by the dual space ${\displaystyle V^{*}}$, which is a subring of the ring of all functions ${\displaystyle V\to k}$. If we fix a basis for V and write ${\displaystyle t_{i}}$ for its dual basis, then k[V] consists of polynomials in ${\displaystyle t_{i}}$.

If k is infinite, then k[V] is the symmetric algebra of the dual space ${\displaystyle V^{*}}$.

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.