linearization
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:
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Method 1. Given any homogeneous polynomial of degree in indeterminates over a commutative scalar ring (scalar simply means that the elements of commute with the indeterminates).
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Step 1
If all indeterminates are linear in , then we are done.
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Step 2
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 .
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Step 3
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.
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Step 1
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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:
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Step 1
If is not linear in and that , replace with a formal linear combination of indeterminates over :
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Step 2
Define a polynomial , the non-commuting free algebra over (generated by the non-commuting indeterminates ) by:
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Step 3
Expand and take the sum of the monomials in whose coefficent is . The result is a linearization of for the indeterminate .
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Step 4
Take the next non-linear indeterminate and start over (with Step 1). Repeat the process until is completely linearized.
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Step 1
Remarks.
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The method of linearization is used often in the studies of Lie algebras, Jordan algebras, PI-algebras and quadratic forms.
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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.
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If is a homogeneous polynomial of degree , then the linearized is a multilinear map in indeterminates.
Examples.
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. Then is a linearization of . In general, if , then the linearization of is
where is the symmetric group on . If in addition all the indeterminates commute with each other and in the ground ring, then the linearization becomes
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. 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
Title | linearization |
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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 |