homogeneous polynomial

homogeneous polynomial

Let

R

be an

associative ring

. A (multivariate)

polynomial

f

over

R

is said to be

homogeneous of degree

r

if it is

expressible

as an

R

-

linear combination

(

http://planetmath.org/LinearCombination

) of

monomials

of degree

r

:

f

⁢

(

x

1

,

…

,

x

n

)

=

∑

i

=

1

m

a

i

⁢

x

1

r

i

⁢

1

⁢

⋯

⁢

x

n

r

i

⁢

n

,

where

r

=

r

i

⁢

1

+

⋯

+

r

i

⁢

n

for all

i

∈

{

1

,

…

,

m

}

and

a

i

∈

R

.

A general

homogeneous polynomial

is also known sometimes as a

polynomial form

. A homogeneous polynomial of degree 1 is called a

linear form

; a homogeneous polynomial of degree 2 is called a

quadratic form

; and a homogeneous polynomial of degree 3 is called a

cubic form

.

Remarks

.

1.

If

f

is a homogeneous polynomial over a ring

R

with

deg

⁡

(

f

)

=

r

, then

f

⁢

(

t

⁢

x

1

,

…

,

t

⁢

x

n

)

=

t

r

⁢

f

⁢

(

x

1

,

…

,

x

n

)

. In fact, a

homogeneous function

that is also a polynomial is a homogeneous polynomial.

2.

Every polynomial

f

over

R

can be expressed uniquely as a finite sum of homogeneous polynomials.
The homogeneous polynomials that make up the polynomial

f

are called the

homogeneous components

of

f

.

3.

If

f

and

g

are homogeneous polynomials of degree

r

and

s

over a domain

R

, then

f

⁢

g

is homogeneous of degree

r

+

s

. From this, one sees that given a domain

R

, the ring

R

⁢

[

𝑿

]

is a

graded ring

, where

𝑿

is a finite set of

indeterminates

. The condition that

R

does not have any

zero divisors

is essential here. As a counterexample, in

ℤ

6

⁢

[

x

,

y

]

, if

f

⁢

(

x

,

y

)

=

2

⁢

x

+

4

⁢

y

and

g

⁢

(

x

,

y

)

=

3

⁢

x

+

3

⁢

y

, then

f

⁢

(

x

,

y

)

⁢

g

⁢

(

x

,

y

)

=

0

.

Examples

•

f

⁢

(

x

,

y

)

=

x

2

+

x

⁢

y

+

y

⁢

x

+

y

2

is a homogeneous polynomial of degree 2. Notice the middle two monomials could be combined into the monomial 2xy if the

variables

are allowed to commute with one another.

•

f

⁢

(

x

)

=

x

3

+

1

is not a homogeneous polynomial.

•

f

⁢

(

x

,

y

,

z

)

=

x

3

+

x

⁢

y

⁢

z

+

z

⁢

y

⁢

z

+

3

⁢

x

⁢

y

2

+

x

2

-

x

⁢

y

+

y

2

+

z

⁢

y

+

z

2

+

x

⁢

z

+

y

+

2

⁢

x

+

6

is a polynomial that is the sum of four homogeneous
polynomials:

x

3

+

x

⁢

y

⁢

z

+

z

⁢

y

⁢

z

+

3

⁢

x

⁢

y

2

(with degree 3),

x

2

-

x

⁢

y

+

y

2

+

z

⁢

y

+

z

2

+

x

⁢

z

(degree = 2),

y

+

2

⁢

x

(degree = 1) and

6

(deg = 0).

•

Every

symmetric polynomial

can be written as a sum of

symmetric

homogeneous polynomials.

Title

homogeneous polynomial

Canonical name

HomogeneousPolynomial1

Date of creation

2013-03-22 14:53:42

Last modified on

2013-03-22 14:53:42

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16R99

Classification

msc 13B25

Classification

msc 16S36

Classification

msc 11E76

Synonym

polynomial form

Related topic

HomogeneousIdeal

Related topic

HomogeneousFunction

Related topic

HomogeneousEquation

Defines

homogeneous component

Defines

cubic form

Defines

linear form

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