Multilinear polynomial

<p>In algebra, a multilinear polynomial is a <a href="/facts/Multivariate_function/cyujR2q2">multivariate</a> <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> that is <a href="/facts/Linear/fChtg3KC">linear</a> (meaning <a href="/facts/Affine_transformation/BD7yDr10">affine</a>) in each of its variables <a href="/facts/Multilinear_map/YosSfBKE">separately</a>, but not necessarily <a href="/facts/Linear_map/Q1PBKNVV">simultaneously</a>. It is a polynomial in which no variable occurs to a power of 
  
    
      
        2
      
    
    {\displaystyle 2}
  
 or higher; that is, each <a href="/facts/Monomial/6VAPCYXJ">monomial</a> is a constant times a product of distinct variables. For example 
  
    
      
        f
        (
        x
        ,
        y
        ,
        z
        )
        =
        3
        x
        y
        +
        2.5
        y
        −
        7
        z
      
    
    {\displaystyle f(x,y,z)=3xy+2.5y-7z}
  
 is a multilinear polynomial of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> 
  
    
      
        2
      
    
    {\displaystyle 2}
  
 (because of the monomial 
  
    
      
        3
        x
        y
      
    
    {\displaystyle 3xy}
  
) whereas 
  
    
      
        f
        (
        x
        ,
        y
        ,
        z
        )
        =
        
          x
          
            2
          
        
        +
        4
        y
      
    
    {\displaystyle f(x,y,z)=x^{2}+4y}
  
 is not. The <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.
</p>

Multilinear polynomial open-in-new

Multilinear polynomial