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theorems on sums of squares (Theorem)
Theorem 1 (Hurwitz Theorem)   Let $ F$ be a field with characteristic not $ 2$. The sum of squares identity of the form
$\displaystyle (x_1^2+\cdots+x_n^2)(y_1^2+\cdots+y_n^2)=z_1^2+\cdots+z_n^2$
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 $ \mathbb{R}$, this theorem is equivalent to the fact that the only normed real division alternative algebra is one of $ \mathbb{R}$, $ \mathbb{C}$, $ \mathbb{H}$, $ \mathbb{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 2 (Pfister's Theorem)   Let $ F$ be a field of characteristic not $ 2$. The sum of squares identity of the form
$\displaystyle (x_1^2+\cdots+x_n^2)(y_1^2+\cdots+y_n^2)=z_1^2+\cdots+z_n^2$
where each $ z_k$ is a rational function of $ x_i$ and $ y_j$ (element of $ F(x_1,\ldots,x_n,y_1,\ldots,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

$\displaystyle (x_1^2+\cdots+x_n^2)(y_1^2+\cdots+y_n^2)=z_1^2+\cdots+z_n^2$
such that each $ z_k$ is a rational function of the $ x_i$ and a linear function of the $ y_j$, or that
$\displaystyle z_k=\sum_{j=1}^{n}r_{kj}y_j$   where $\displaystyle r_{kj}\in F(x_1,\ldots,x_n).$
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\in F(x_1,\ldots,x_n,y_1,\ldots,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 $ \mathbb{R}$, we have the following generalization, also due to Pfister:

Theorem 3   If $ f\in\mathbb{R}(X_1,\ldots,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.

Bibliography

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).



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See Also: Mazur's structure theorem, sums of two squares, stufe of a field, Cayley-Dickson construction, composition algebra, octonion

Other names:  Pfister theorem
Also defines:  Hurwitz theorem, Pfister's theorem, Hilbert's 17th Problem
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Cross-references: rationals, indeterminates, positive semidefinite, function, rational function, composition algebras, characterization, algebras, norm, alternative algebra, division, real, equivalent, ground field, iff, coefficients, bilinear, identity, squares, sum, characteristic, field
There is 1 reference to this entry.

This is version 11 of theorems on sums of squares, born on 2005-02-25, modified 2006-10-01.
Object id is 6830, canonical name is TheoremsOnSumsOfSquares.
Accessed 7475 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
 16D60 (Associative rings and algebras :: Modules, bimodules and ideals :: Simple and semisimple modules, primitive rings and ideals)
 12D15 (Field theory and polynomials :: Real and complex fields :: Fields related with sums of squares )
 11E25 (Number theory :: Forms and linear algebraic groups :: Sums of squares and representations by other particular quadratic forms)
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second attempt: when to use possesive case??? by mathforever on 2005-02-26 06:51:47
Hello everybody!

I have posted a message a while ago, but no one replied on it, may be because it was exteremely busy time, so I decided to post it again:

------------------------------------------
Is there any rule which says what is right

"Planck's constant" or "Planck constant" (the same with Euler)
"Green's function" or "Green function"
"Poincare's inequality" or "Poincare inequality"
and so on...

I saw in some places with "'s" in some places without. May be both variants are acceptable, but may be variants with "'s" mean something different then without?

It would be nice to have some clear explanation. Thanks in advance :)
------------------------------------------

The reason to post it here (near this entry), is that there are two names:

Hurwitz theorem, Pfister's theorem

one with 's and one without, and thus the question has applied nature: why "Hurwitz thrm" is without 's and "Pfister's thrm" is with 's?????

Thanks in advance!
Serg.

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