Ring of polynomial functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the ring of polynomial functions on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> k gives a coordinate-free analog of a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a>. It is denoted by k[V]. If V is <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite dimensional</a> and is viewed as an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a>, then k[V] is precisely the <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> of V.
The explicit definition of the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> can be given as follows. Given a polynomial ring 
 
 
 
 k
 [
 
 t
 
 1
 
 
 ,
 …
 ,
 
 t
 
 n
 
 
 ]
 
 
 {\displaystyle k[t_{1},\dots ,t_{n}]}
 
, we can view 
 
 
 
 
 t
 
 i
 
 
 
 
 {\displaystyle t_{i}}
 
 as a coordinate function on 
 
 
 
 
 k
 
 n
 
 
 
 
 {\displaystyle k^{n}}
 
; i.e., 
 
 
 
 
 t
 
 i
 
 
 (
 x
 )
 =
 
 x
 
 i
 
 
 
 
 {\displaystyle t_{i}(x)=x_{i}}
 
 where 
 
 
 
 x
 =
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 .
 
 
 {\displaystyle x=(x_{1},\dots ,x_{n}).}
 
 This suggests the following:[how?] given a vector space V, let k[V] be the <a href="/facts/Associative_algebra/DRfoF6KV">commutative k-algebra</a> generated by the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> 
 
 
 
 
 V
 
 ∗
 
 
 
 
 {\displaystyle V^{*}}
 
, which is a <a href="/facts/Subring/M0perybt">subring</a> of the ring of all <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> 
 
 
 
 V
 →
 k
 
 
 {\displaystyle V\to k}
 
. If we fix a <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> for V and write 
 
 
 
 
 t
 
 i
 
 
 
 
 {\displaystyle t_{i}}
 
 for its dual basis, then k[V] consists of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in 
 
 
 
 
 t
 
 i
 
 
 
 
 {\displaystyle t_{i}}
 
.
If k is infinite, then k[V] is the <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a> of the dual space 
 
 
 
 
 V
 
 ∗
 
 
 
 
 {\displaystyle V^{*}}
 
.
In applications, one also defines k[V] when V is defined over some <a href="/facts/Field_extension/L7yIDtyK">subfield</a> of k (e.g., k is the <a href="/facts/Complex_number/2w7mqI7e">complex</a> field and V is a <a href="/facts/Real_number/R02gw5Pb">real</a> vector space.) The same definition still applies.
Throughout the article, for simplicity, the base field k is assumed to be infinite.

Ring of polynomial functions open-in-new

Ring of polynomial functions