Linearization is the process of reducing a homogeneous polynomialMathworldPlanetmath into a multilinear map over a commutative ring. There are in general two ways of doing this:

  • Method 1. Given any homogeneous polynomial f of degree n in m indeterminates over a commutativePlanetmathPlanetmathPlanetmathPlanetmath scalar ring R (scalar simply means that the elements of R commute with the indeterminates).

    1. Step 1

      If all indeterminates are linear in f, then we are done.

    2. Step 2

      Otherwise, pick an indeterminate x such that x is not linear in f. Without loss of generality, write f=f(x,X), where X is the set of indeterminates in f excluding x. Define g(x1,x2,X):=f(x1+x2,X)-f(x1,X)-f(x2,X). Then g is a homogeneous polynomial of degree n in m+1 indeterminates. However, the highest degree of x1,x2 is n-1, one less that of x.

    3. Step 3

      Repeat the process, starting with Step 1, for the homogeneous polynomial g. Continue until the set X of indeterminates is enlarged to one X such that each xX is linear.

  • Method 2. This method applies only to homogeneous polynomials that are also homogeneousPlanetmathPlanetmath in each indeterminate, when the other indeterminates are held constant, i.e., f(tx,X)=tnf(x,X) for some n and any tR. Note that if all of the indeterminates in f commute with each other, then f is essentially a monomialMathworldPlanetmath. So this method is particularly useful when indeterminates are non-commuting. If this is the case, then we use the following algorithmMathworldPlanetmath:

    1. Step 1

      If x is not linear in f and that f(tx,X)=tnf(x,X), replace x with a formal linear combinationMathworldPlanetmath of n indeterminates over R:

      r1x1++rnxn, where riR.
    2. Step 2

      Define a polynomialMathworldPlanetmath gRx1,,xn, the non-commuting free algebraMathworldPlanetmath over R (generated by the non-commuting indeterminates xi) by:

    3. Step 3

      Expand g and take the sum of the monomials in g whose coefficent is r1rn. The result is a linearization of f for the indeterminate x.

    4. Step 4

      Take the next non-linear indeterminate and start over (with Step 1). Repeat the process until f is completely linearized.


  1. 1.

    The method of linearization is used often in the studies of Lie algebras, Jordan algebrasMathworldPlanetmathPlanetmath, PI-algebras and quadratic formsMathworldPlanetmath.

  2. 2.

    If the characteristic of scalar ring R is 0 and f is a monomial in one indeterminate, we can recover f back from its linearization by setting all of its indeterminates to a single indeterminate x and dividing the resulting polynomial by n!:


    Please see the first example below.

  3. 3.

    If f is a homogeneous polynomial of degree n, then the linearized f is a multilinear map in n indeterminates.


  • f(x)=x2. Then f(x1+x2)-f(x1)-f(x2)=x1x2+x2x1 is a linearization of x2. In general, if f(x)=xn, then the linearization of f is


    where Sn is the symmetric groupPlanetmathPlanetmath on {1,,n}. If in addition all the indeterminates commute with each other and n!0 in the ground ring, then the linearization becomes

  • f(x,y)=x3y2+xyxyx. Since f(tx,y)=t3f(x,y) and f(x,ty)=t2f(x,y), f is homogeneous over x and y separately, and thus we can linearize f. First, collect all the monomials having coefficient abc in (ax1+bx2+cx3,y), we get


    where i,j,k1,2,3 and (i-j)(j-k)(k-i)0. Repeat this for y and we have

Title linearization
Canonical name Linearization
Date of creation 2013-03-22 14:53:52
Last modified on 2013-03-22 14:53:52
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Classification msc 15A69
Classification msc 16R99
Classification msc 17A99
Synonym polarization
Defines linearized