# 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 $S = R[t_1, \dots, t_n]$, 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.