Linearization is the process of reducing a homogeneous polynomial 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 commutative scalar ring R (scalar simply means that the elements of Rcommute with the indeterminates).
  

  Step 1

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

  Step 2

  Otherwise, pick an indeterminate x such that x is not linear in f . Without loss of generality, write f=f(xX) , where X is the set of indeterminates in fexcluding x . Define g(x1x2X):=f(x1+x2X)−f(x1X)−f(x2X) . Then g is a homogeneous polynomial of degree n in m+1 indeterminates. However, the highest degree of x1x2 is n−1 , one less that of x .

  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 eachxX is linear.



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

  Step 1

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

  r1x1++rnxn, where riR

  Step 2

  Define a polynomial gRx1xn , the non-commuting free algebra over R (generated by the non-commuting indeterminates xi ) by:

  g(x1xn):=f(r1x1++rnxn)

  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 .

  Step 4

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

Remarks.

1. The method of linearization is used often in the studies of Lie algebras, Jordan algebras, PI-algebras and quadratic forms.
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! :
   f(x)=1n!linearization(f)(xx)
   Please see the first example below.
3. If f is a homogeneous polynomial of degree n , then the linearized f is a multilinear map in n indeterminates.

Examples.

• 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
  Snx(1)x(n)=Snni=1x(i) 
  where Sn is the symmetric group on 1n . If in addition all the indeterminates commute with each other and n!=0 in the ground ring, then the linearization becomes
  n!x1xn=ni=1ixi 

polarization

polarization of a polynomial

The fundamental ideas are as follows. Let f(u) be a polynomial in n variables u = (u₁, u₂, ..., u_n). Suppose that f is homogeneous of degree d, which means that

f(t u) = t^d f(u) for all t.

Let u⁽¹⁾, u⁽²⁾, ..., u^(d) be a collection of indeterminates with u⁽ⁱ⁾ = (u₁⁽ⁱ⁾, u₂⁽ⁱ⁾, ..., u_n⁽ⁱ⁾), so that there are dn variables altogether. The polar form of f is a polynomial

F(u⁽¹⁾, u⁽²⁾, ..., u^(d))

which is linear separately in each u⁽ⁱ⁾ (i.e., F is multilinear), symmetric in the u⁽ⁱ⁾, and such that

F(u,u, ..., u)=f(u).

The polar form of f is given by the following construction

In other words, F is a constant multiple of the coefficient of λ₁ λ₂...λ_d in the expansion of f(λ₁u⁽¹⁾ + ... + λ_du^(d)).