theorems on sums of squares

theorems on sums of squares

Theorem

().

Let

F

be a field with

characteristic

not

2

. The sum of squares

identity

of the form

(

x

1

2

+

⋯

+

x

n

2

)

⁢

(

y

1

2

+

⋯

+

y

n

2

)

=

z

1

2

+

⋯

+

z

n

2

where each

z

k

is

bilinear

over

x

i

and

y

j

(with

coefficients

in

F

), is possible iff

n

=

1

,

2

,

4

,

8

.

Remarks

.

1.

When the

ground field

is

ℝ

, this

theorem

is

equivalent

to the fact that the only normed real

division

alternative algebra

is one of

ℝ

,

ℂ

,

ℍ

,

𝕆

, as one observes that the sums of squares can be interpreted as the square of the norm defined for each of the above

algebras

.

2.

An equivalent

characterization

is that the above four mentioned algebras are the only real

composition algebras

.

A

generalization

of the above is the following:

Theorem

(Pfister’s Theorem).

Let

F

be a field of
characteristic not

2

. The sum of squares identity of the form

(

x

1

2

+

⋯

+

x

n

2

)

⁢

(

y

1

2

+

⋯

+

y

n

2

)

=

z

1

2

+

⋯

+

z

n

2

where each

z

k

is a

rational function

of

x

i

and

y

j

(element
of

F

⁢

(

x

1

,

…

,

x

n

,

y

1

,

…

,

y

n

)

), is possible iff

n

is a
power of

2

.

Remark

. The form of Pfister’s theorem is stated in a way
so as to mirror the form of

Hurwitz theorem

. In fact, Pfister
proved the following: if

F

is a field and

n

is a power of 2,
then there exists a sum of squares identity of the form

(

x

1

2

+

⋯

+

x

n

2

)

⁢

(

y

1

2

+

⋯

+

y

n

2

)

=

z

1

2

+

⋯

+

z

n

2

such
that each

z

k

is a rational function of the

x

i

and a

linear
function

of the

y

j

, or that

z

k

=

∑

j

=

1

n

r

k

⁢

j

⁢

y

j

where

⁢

r

k

⁢

j

∈

F

⁢

(

x

1

,

…

,

x

n

)

.

Conversely

, if

n

is not a power of

2

, then
there exists a field

F

such that the above sum of square identity
does not hold for

any

z

i

∈

F

⁢

(

x

1

,

…

,

x

n

,

y

1

,

…

,

y

n

)

. Notice that

z

i

is no longer
required to be a linear function of the

y

j

anymore.

When

F

is the

field of reals

ℝ

, we have the following
generalization, also due to Pfister:

Theorem

.

If

f

∈

R

⁢

(

X

1

,

…

,

X

n

)

is

positive semidefinite

, then

f

can be written as a sum of

2

n

squares.

The above theorem is very closely related to Hilbert’s 17th Problem:

Hilbert’s 17th Problem.

Whether it is possible, to
write a positive semidefinite rational function in

n

indeterminates

over the reals, as a sum of squares of rational
functions in

n

indeterminates over the reals?

The answer is yes, and it was proved by Emil Artin in 1927.
Additionally, Artin showed that the answer is also yes if the reals
were replaced by the

rationals

.

References

1

A. Hurwitz,

Über die Komposition der quadratishen Formen von beliebig vielen Variabeln

, Nachrichten von der Königlichen Gesellschaft der Wissenschaften in Göttingen (1898).

2

A. Pfister,

Zur Darstellung definiter Funktionen als Summe von Quadraten

, Inventiones Mathematicae (1967).

3

A. R. Rajwade,

Squares

, Cambridge University Press (1993).

4

J. Conway, D. A. Smith,

On

Quaternions

and Octonions

, A K Peters, LTD. (2002).

Title

theorems on sums of squares

Canonical name

TheoremsOnSumsOfSquares

Date of creation

2013-03-22 15:06:04

Last modified on

2013-03-22 15:06:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 12D15

Classification

msc 16D60

Classification

msc 15A63

Classification

msc 11E25

Synonym

Pfister theorem

Related topic

MazursStructureTheorem

Related topic

SumsOfTwoSquares

Related topic

StufeOfAField

Related topic

CayleyDicksonConstruction

Related topic

CompositionAlgebra

Related topic

Octonion

Defines

Hurwitz theorem

Defines

Pfister’s theorem

Defines

Hilbert’s 17th Problem

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