linearization

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:

•

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

R

commute 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

⁢

(

x

,

X

)

, where

X

is the set of indeterminates in

f

excluding

x

. Define

g

⁢

(

x

1

,

x

2

,

X

)

:=

f

⁢

(

x

1

+

x

2

,

X

)

-

f

⁢

(

x

1

,

X

)

-

f

⁢

(

x

2

,

X

)

. Then

g

is a homogeneous polynomial of degree

n

in

m

+

1

indeterminates. However, the highest degree of

x

1

,

x

2

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 each

x

∈

X

′

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

⁢

(

t

⁢

x

,

X

)

=

t

n

⁢

f

⁢

(

x

,

X

)

for some

n

∈

ℕ

and
any

t

∈

R

. Note that if all of the indeterminates in

f

commute with each other, then

f

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

:

Step 1

If

x

is not linear in

f

and that

f

⁢

(

t

⁢

x

,

X

)

=

t

n

⁢

f

⁢

(

x

,

X

)

, replace

x

with a formal

linear combination

of

n

indeterminates over

R

:

r

1

⁢

x

1

+

⋯

+

r

n

⁢

x

n

⁢

, where

⁢

r

i

∈

R

.

Step 2

Define a

polynomial

g

∈

R

⁢

⟨

x

1

,

…

,

x

n

⟩

, the non-commuting

free algebra

over

R

(generated by the non-commuting indeterminates

x

i

) by:

g

⁢

(

x

1

,

…

,

x

n

)

:=

f

⁢

(

r

1

⁢

x

1

+

⋯

+

r

n

⁢

x

n

)

.

Step 3

Expand

g

and take the sum of the monomials in

g

whose coefficent is

r

1

⁢

⋯

⁢

r

n

. 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

)

=

1

n

!

⁢

linearization

⁡

(

f

)

⁢

(

x

,

…

,

x

)

.

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

)

=

x

2

. Then

f

⁢

(

x

1

+

x

2

)

-

f

⁢

(

x

1

)

-

f

⁢

(

x

2

)

=

x

1

⁢

x

2

+

x

2

⁢

x

1

is a linearization of

x

2

. In general, if

f

⁢

(

x

)

=

x

n

,
then the linearization of

f

is

∑

σ

∈

S

n

x

σ

⁢

(

1

)

⁢

⋯

⁢

x

σ

⁢

(

n

)

=

∑

σ

∈

S

n

∏

i

=

1

n

x

σ

⁢

(

i

)

,

where

S

n

is the

symmetric group

on

{

1

,

…

,

n

}

.
If in

addition

all the indeterminates commute with each other and

n

!

≠

0

in the

ground ring

, then the linearization
becomes

n

!

⁢

x

1

⁢

⋯

⁢

x

n

=

∏

i

=

1

n

i

⁢

x

i

.

•

f

⁢

(

x

,

y

)

=

x

3

⁢

y

2

+

x

⁢

y

⁢

x

⁢

y

⁢

x

. Since

f

⁢

(

t

⁢

x

,

y

)

=

t

3

⁢

f

⁢

(

x

,

y

)

and

f

⁢

(

x

,

t

⁢

y

)

=

t

2

⁢

f

⁢

(

x

,

y

)

,

f

is homogeneous over

x

and

y

separately, and thus we can linearize

f

. First, collect all the monomials having

coefficient

a

⁢

b

⁢

c

in

(

a

⁢

x

1

+

b

⁢

x

2

+

c

⁢

x

3

,

y

)

, we get

g

⁢

(

x

1

,

x

2

,

x

3

,

y

)

:=

∑

x

i

⁢

x

j

⁢

x

k

⁢

y

2

+

x

i

⁢

y

⁢

x

j

⁢

y

⁢

x

k

,

where

i

,

j

,

k

∈

1

,

2

,

3

and

(

i

-

j

)

⁢

(

j

-

k

)

⁢

(

k

-

i

)

≠

0

. Repeat this for

y

and we have

h

⁢

(

x

1

,

x

2

,

x

3

,

y

1

,

y

2

)

:=

∑

x

i

⁢

x

j

⁢

x

k

⁢

(

y

1

⁢

y

2

+

y

2

⁢

y

1

)

+

(

x

i

⁢

y

1

⁢

x

j

⁢

y

2

⁢

x

k

+

x

i

⁢

y

2

⁢

x

j

⁢

y

1

⁢

x

k

)

.

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

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