proof of fundamental theorem of symmetric polynomials

proof of fundamental theorem of symmetric polynomials

Let

P

:=

P

⁢

(

x

1

,

x

2

,

…

,

x

n

)

be an arbitrary

symmetric polynomial

in

x

1

,

x

2

,

…

,

x

n

.  We can assume that

P

is

homogeneous

(

http://planetmath.org/

Polynomial

), because if

P

=

P

1

+

P

2

+

…

+

P

m

where each

P

i

is homogeneous and if the

theorem

(

http://planetmath.org/FundamentalTheoremOfSymmetricPolynomials

) is true for each

P

i

, it is evidently true for the sum

P

, too.

Let the degree (

http://planetmath.org/Polynomial

) of

P

be

d

.  For any two terms

M

:=

c

μ

⁢

x

1

μ

1

⁢

x

2

μ

2

⁢

⋯

⁢

x

n

μ

n

,

N

:=

c

ν

⁢

x

1

ν

1

⁢

x

2

ν

2

⁢

⋯

⁢

x

n

ν

n

of

P

, if the first of the

differences

μ

1

-

ν

1

,

μ

2

-

ν

2

,

…

,

μ

n

-

ν

n

,

which differs from 0, is

positive

, we say that

M

is

higher

than

N

.  Since, of cource, the terms of

P

have been merged, always one of two arbitrary terms is higher than the other.  The higherness is obviously

transitive

(

http://planetmath.org/Transitive3

).  Thus there is a certain

highest

term

A

:=

c

α

⁢

x

1

α

1

⁢

x

2

α

2

⁢

⋯

⁢

x

n

α

n

in

P

.  Then we have

α

1

+

α

2

+

…

+

α

n

=

d

,

α

1

≥

α

2

≥

…

≥

α

n

.

In fact, if e.g.

α

2

>

α

1

,  then the term

c

α

⁢

x

2

α

1

⁢

x

1

α

2

⁢

x

3

α

3

⁢

⋯

⁢

x

n

α

n

=

c

α

⁢

x

1

α

2

⁢

x

2

α

1

⁢

x

3

α

3

⁢

⋯

⁢

x

n

α

n

,

which is obtained from

A

by changing

x

1

and

x

2

with each other, would be higher than

A

.

For proving the fundamental theorem (

http://planetmath.org/FundamentalTheoremOfSymmetricPolynomials

), we form now the

homogeneous polynomial

Q

α

:=

c

α

⁢

p

1

i

1

⁢

p

2

i

2

⁢

⋯

⁢

p

n

i

n

and we will show that the exponents (

http://planetmath.org/

Exponentiation

)

i

j

can be determined such that the highest term in

Q

α

is same as in

P

.

It is easily seen that the highest term of a

product

of homogeneous symmetric polynomials is equal to the product of the highest terms of the factors.  Since the highest term of

p

1

is

x

1

,

p

2

is

x

1

⁢

x

2

,

⋯

⋯

p

n

is

x

1

⁢

x

2

⁢

⋯

⁢

x

n

,

therefore the highest term of

p

1

i

1

is

x

1

i

1

,

p

2

i

2

is

x

1

i

2

⁢

x

2

i

2

,

⋯

⋯

p

n

i

n

is

x

1

i

n

⁢

x

2

i

n

⁢

⋯

⁢

x

n

i

n

and thus the highest term of

Q

α

is

c

α

⁢

x

1

i

1

+

i

2

+

⋯

+

i

n

⁢

x

2

i

2

+

⋯

+

i

n

⁢

⋯

⁢

x

n

i

n

.

This term coincides with the highest term of

P

, when one determines the numbers

i

j

from the equations

{

i

1

+

i

2

+

…

+

i

n

=

α

1

i

2

+

…

+

i

n

=

α

2

⋯

⋯

i

n

=

α

n

.

Subtracting here the second equation from the first, the third equation from the second and so on, the result is

i

1

=

α

1

-

α

2

,

i

2

=

α

2

-

α

3

,

…

,

i

n

-

1

=

α

n

-

1

-

α

n

,

i

n

=

α

n

,

which are

nonnegative integers

.  Hence we get the homogeneous symmetric polynomial

Q

α

=

c

α

⁢

p

1

α

1

-

α

2

⁢

p

2

α

2

-

α

3

⁢

⋯

⁢

p

n

-

1

α

n

-

1

-

α

n

⁢

p

n

α

n

having the same highest term as

P

, and consequently the difference

P

-

Q

α

:=

P

α

is a homogeneous symmetric polynomial of degree

d

having the highest term lower than in

P

.  If then

c

β

⁢

x

1

β

1

⁢

x

2

β

2

⁢

⋯

⁢

x

n

β

n

is the highest term of

P

α

and one denotes

Q

β

:=

c

β

⁢

p

1

β

1

-

β

2

⁢

p

2

β

2

-

β

3

⁢

⋯

⁢

p

n

-

1

β

n

-

1

-

β

n

⁢

p

n

β

n

,

one infers as above that the difference

P

α

-

Q

β

:=

P

β

is a homogeneous symmetric polynomial of degree

d

having the highest term lower than in

P

α

.  Continuing similarly, one finally (after a finite amount of steps) shall come to a difference which is equal to 0.  Accordingly one obtains

P

=

Q

α

+

Q

β

+

…

+

Q

ω

:=

Q

⁢

(

p

1

,

p

2

,

…

,

p

n

)

.

The degree of

Q

α

with respect to the

elementary symmetric polynomials

is

(

α

1

-

α

2

)

+

(

α

2

-

α

3

)

+

…

+

(

α

n

-

1

-

α

n

)

+

α

n

=

α

1

.

Similarly, the degree of

Q

β

is

β

1

which is

≤

α

1

; thus one infers that the degree of

Q

is equal to

α

1

.  This number is also the degree of the highest term of

P

and as well the degree of

P

itself, with respect to

x

1

.

The preceding construction implies immediately that the

coefficients

of

G

are elements of the ring determined by the coefficients of

P

.  We have still to prove the uniqueness of

Q

.  Let’s make the antithesis that

P

may be represented also by another polynomial in

p

1

,

p

2

,

…

,

p

n

which differs from

Q

.  Forming the difference of it and

Q

we get an equation of the form

0

=

∑

i

g

i

⁢

p

1

i

1

⁢

p

2

i

2

⁢

⋯

⁢

p

n

i

n

,

where the coefficients are distinct from zero.  The equation becomes identical if one expresses

p

1

,

p

2

,

…

,

p

n

in it via the

indeterminates

x

1

,

x

2

,

…

,

x

n

.  The general term of the right hand side of the equation is a homogeneous symmetric polynomial in those indeterminates; if its highest term is

g

i

⁢

p

1

λ

1

⁢

p

2

λ

2

⁢

⋯

⁢

p

n

λ

n

,

one infers as before that

i

1

=

λ

1

-

λ

2

,

i

2

=

λ

2

-

λ

3

,

…

,

i

n

-

1

=

λ

n

-

1

-

λ

n

,

i

n

=

λ

n

.

Thus, distinct addends of the sum cannot have equal highest terms.  It means that the highest term of the sum appears only in one of the addends of the sum.  This is, however, impossible, because after the substitution of

x

i

s the equation would not be identical.  Consequently, the antithesis is wrong and the whole

fundamental theorem of symmetric polynomials

has been proved.

References

1

K. Väisälä:

Lukuteorian ja korkeamman algebran alkeet

.  Tiedekirjasto No. 17.   Kustannusosakeyhtiö Otava, Helsinki (1950).

Title

proof of fundamental theorem of symmetric polynomials

Canonical name

ProofOfFundamentalTheoremOfSymmetricPolynomials

Date of creation

2016-02-22 13:29:31

Last modified on

2016-02-22 13:29:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Proof

Classification

msc 12F10

Classification

msc 13B25

Synonym

proof of fundamental theorem of symmetric functions

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