If all indeterminates are linear in , then we are done.
Otherwise, pick an indeterminate such that is not linear in . Without loss of generality, write , where is the set of indeterminates in excluding . Define . Then is a homogeneous polynomial of degree in indeterminates. However, the highest degree of is , one less that of .
Repeat the process, starting with Step 1, for the homogeneous polynomial . Continue until the set of indeterminates is enlarged to one such that each is linear.
- Step 1
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., for some and any . Note that if all of the indeterminates in commute with each other, then is essentially a monomial. So this method is particularly useful when indeterminates are non-commuting. If this is the case, then we use the following algorithm:
If is not linear in and that , replace with a formal linear combination of indeterminates over :
- Step 2
Expand and take the sum of the monomials in whose coefficent is . The result is a linearization of for the indeterminate .
Take the next non-linear indeterminate and start over (with Step 1). Repeat the process until is completely linearized.
- Step 1
If the characteristic of scalar ring is 0 and is a monomial in one indeterminate, we can recover back from its linearization by setting all of its indeterminates to a single indeterminate and dividing the resulting polynomial by :
Please see the first example below.
If is a homogeneous polynomial of degree , then the linearized is a multilinear map in indeterminates.
. Since and , is homogeneous over and separately, and thus we can linearize . First, collect all the monomials having coefficient in , we get
where and . Repeat this for and we have
|Date of creation||2013-03-22 14:53:52|
|Last modified on||2013-03-22 14:53:52|
|Last modified by||CWoo (3771)|