Naringenin 7-O-methyltransferase

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 $S=R[t_{1},\dots ,t_{n}]$, then, as the notation suggests, we can view $t_{i}$ as coordinate functions on $R^{n}$: $t_{i}(x)=x_{i}$ when $x=(x_{1},\dots ,x_{n}).$ 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 $V^{*}$ of the ring of functions $V\to k$. If we fix a basis for V and write $t_{i}$ for its dual basis, then S consists of polynomials in $t_{i}$; 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.

See also

• Algebraic geometry of projective spaces
• Symmetric algebra
• Zariski tangent space