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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 <math>S = R[t_1, \dots, t_n]</math>, then, as the notation suggests, we can view <math>t_i</math> as coordinate functions on <math>R^n</math>: <math>t_i(x) = x_i</math> when <math>x = (x_1, \dots, x_n).</math> 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 <math>V^*</math> of the ring of functions <math>V \to k</math>. If we fix a basis for ''V'' and write <math>t_i</math> for its dual basis, then ''S'' consists of polynomials in <math>t_i</math>; ''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]]

[[Category:Polynomials]]
[[Category:Ring theory]]

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