Linearised polynomial

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a linearised polynomial (or q-polynomial) is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> for which the exponents of all the constituent <a href="/facts/Monomial/6VAPCYXJ">monomials</a> are <a href="/facts/Power_(mathematics)/pac6QY2g">powers</a> of q and the <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> come from some <a href="/facts/Extension_field/L7yIDtyK">extension field</a> of the <a href="/facts/Finite_field/UkMGJo3m">finite field</a> of order q.
We write a typical example as

L
        (
        x
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            n
          
        
        
          a
          
            i
          
        
        
          x
          
            
              q
              
                i
              
            
          
        
        ,
      
    
    {\displaystyle L(x)=\sum _{i=0}^{n}a_{i}x^{q^{i}},}

where each 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 is in 
 
 
 
 
 F
 
 
 q
 
 m
 
 
 
 
 (
 =
 GF
 ⁡
 (
 
 q
 
 m
 
 
 )
 )
 
 
 {\displaystyle F_{q^{m}}(=\operatorname {GF} (q^{m}))}
 
 for some fixed positive <a href="/facts/Integer/PUwwrawx">integer</a> 
 
 
 
 m
 
 
 {\displaystyle m}
 
. 
This special class of polynomials is important from both a theoretical and an applications viewpoint. The highly structured nature of their <a href="/facts/Root_of_a_function/mlK3dhoT">roots</a> makes these roots easy to determine.

Linearised polynomial open-in-new

Linearised polynomial