Polynomial mapping

In <a href="/facts/Abstract_algebra/YNNE1CNz">algebra</a>, a polynomial map or polynomial mapping 
 
 
 
 P
 :
 V
 →
 W
 
 
 {\displaystyle P:V\to W}
 
 between <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over an infinite <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> k is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> in <a href="/facts/Linear_functional/oGh7Asrz">linear functionals</a> with coefficients in k; i.e., it can be written as

P
        (
        v
        )
        =
        
          ∑
          
            
              i
              
                1
              
            
            ,
            …
            ,
            
              i
              
                n
              
            
          
        
        
          λ
          
            
              i
              
                1
              
            
          
        
        (
        v
        )
        ⋯
        
          λ
          
            
              i
              
                n
              
            
          
        
        (
        v
        )
        
          w
          
            
              i
              
                1
              
            
            ,
            …
            ,
            
              i
              
                n
              
            
          
        
      
    
    {\displaystyle P(v)=\sum _{i_{1},\dots ,i_{n}}\lambda _{i_{1}}(v)\cdots \lambda _{i_{n}}(v)w_{i_{1},\dots ,i_{n}}}

where the 
 
 
 
 
 λ
 
 
 i
 
 j
 
 
 
 
 :
 V
 →
 k
 
 
 {\displaystyle \lambda _{i_{j}}:V\to k}
 
 are linear functionals and the 
 
 
 
 
 w
 
 
 i
 
 1
 
 
 ,
 …
 ,
 
 i
 
 n
 
 
 
 
 
 
 {\displaystyle w_{i_{1},\dots ,i_{n}}}
 
 are vectors in W. For example, if 
 
 
 
 W
 =
 
 k
 
 m
 
 
 
 
 {\displaystyle W=k^{m}}
 
, then a polynomial mapping can be expressed as 
 
 
 
 P
 (
 v
 )
 =
 (
 
 P
 
 1
 
 
 (
 v
 )
 ,
 …
 ,
 
 P
 
 m
 
 
 (
 v
 )
 )
 
 
 {\displaystyle P(v)=(P_{1}(v),\dots ,P_{m}(v))}
 
 where the 
 
 
 
 
 P
 
 i
 
 
 
 
 {\displaystyle P_{i}}
 
 are (scalar-valued) <a href="/facts/Polynomial_function/Lzak8VVx">polynomial functions</a> on V. (The abstract definition has an advantage that the map is manifestly free of a choice of basis.)
When V, W are <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> vector spaces and are viewed as <a href="/facts/Algebraic_varieties/Vk7f97vn">algebraic varieties</a>, then a polynomial mapping is precisely a <a href="/facts/Morphism_of_algebraic_varieties/5jBzIgps">morphism of algebraic varieties</a>.
One fundamental outstanding question regarding polynomial mappings is the <a href="/facts/Jacobian_conjecture/J2NbRNk4">Jacobian conjecture</a>, which concerns the sufficiency of a polynomial mapping to be invertible.

Polynomial mapping open-in-new

Polynomial mapping