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Hometheorems on sums of squares

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# theorems on sums of squares

###### Theorem (Hurwitz Theorem).

Let $F$ be a field with characteristic not $2$. The sum of squares identity of the form

$(x_{1}^{2}+\cdots+x_{n}^{2})(y_{1}^{2}+\cdots+y_{n}^{2})=z_{1}^{2}+\cdots+z_{n% }^{2}$ |

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 (Pfister’s Theorem).

Let $F$ be a field of characteristic not $2$. The sum of squares identity of the form

$(x_{1}^{2}+\cdots+x_{n}^{2})(y_{1}^{2}+\cdots+y_{n}^{2})=z_{1}^{2}+\cdots+z_{n% }^{2}$ |

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

$(x_{1}^{2}+\cdots+x_{n}^{2})(y_{1}^{2}+\cdots+y_{n}^{2})=z_{1}^{2}+\cdots+z_{n% }^{2}$ |

such that each $z_{k}$ is a rational function of the $x_{i}$ and a linear function of the $y_{j}$, or that

$z_{k}=\sum_{{j=1}}^{{n}}r_{{kj}}y_{j}\qquad\mbox{where }r_{{kj}}\in F(x_{1},% \ldots,x_{n}).$ |

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.

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.

# 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).

## Mathematics Subject Classification

12D15*no label found*16D60

*no label found*15A63

*no label found*11E25

*no label found*

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## Comments

## second attempt: when to use possesive case???

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.

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: second attempt: when to use possesive case???

It should link correctly either way. I'd suggest going with whatever you think the most common usage is. My guess is that this will usually be the possessive form.

apk

## Re: second attempt: when to use possesive case???

I have seen that Hurwitz theorem is used most often, instead of Hurwitz's theorem. Perhaps it is because the z at the end of the name is similar sounding to s. On the other hand, I have never seen Pfister's theorem being stated as Pfister theorem.

Chi

## Re: second attempt: when to use possesive case???

It's impossible to pronounce the genitive marker "s" in the names ending in a sibilant phoneme as in Hurwitz, Weierstrass, Bush. Therefore the "s" letter must not be written in such names, but if one wants to denote the genitive of them then write e.g. Hurwitz' theorem. Otherwise the "s" letter may be written if it's an established, standard procedure.

Jussi

## Re: second attempt: when to use possesive case???

Thanks for replies!

I would like then to ask: does possesiveness affect the meaning of the notion? Originally I submitted a correction to the entry "Simmon's line" for forgetting 's in the phrase like "the line *** is a Simson line for triangle ***"

http://planetmath.org/?op=getobj&from=corrections&id=5786

and I got a reply like 'a' makes it possible to write "Simson line" without 's. So, is it really true that the absence of 's make the notion more general meaning...

I am not sure, but it seems to me now that things like "Simson line", or "Green function", or "Prandtl number" are NOT some fixed 'line', 'function', 'number' and they just have the names of Simson, Green and Prandtl. And that's why one shouldn't use posseive case here.

On the other hand "Fermat's theorem", "Zorn's lemma", "Planck's constant" are some concrete "theorem", "lemma" and "constant" and they actually BELONG to the persons who invented them and thus possesive case should be used here.

What do you think about this?

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: second attempt: when to use possesive case???

ok I think I misunderstood your correction then, I'm sorry

I thought you were complaining about the "THE"

as in "THE line", because there are many of them, not a single one, that's why I changed it to "A" and closed the correction.

Again, It was me who misunderstood the correction

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: second attempt: when to use possesive case???

> ok I think I misunderstood your correction then

Wait, wait! I am not so sure that you misunderstood it ;).

For everyone to understand what we are talking about here is the original sentence which I thought has some problems:

"In the picture, the line passing through $U,V,W$ is the Simson line for $\triangle ABC$."

So, as already said "Simson line" is NOT some fixed line but rather a line determined by some things (see the entry). Thus "the Simson line for triangle..." sounds like triangle has ONE Simson line, which is not the case, so it is better to use "a Simson line".

But what we are talking about now, is another thing: when to use 's? According to what I wrote in the previous post "Simson line" should be written everywhere WITHOUT 's. The big question is of cource, whether what I wrote is actually correct ;). Have you really seen the expression "Simson's line"?

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: second attempt: when to use possesive case???

yes

THAT was the line

yet when you submited your correction I saw:

You probably forgot 's in "line passing through $U,V,W$ is a Simson line for $\triangle ABC$".

So I noticed you had changed THE by A in the text of your correction

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: second attempt: when to use possesive case???

ok I think you had two points to mak in the correction, yet I failed to see one of them, I truly never understood that you were talking about posesives

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: second attempt: when to use possesive case???

But anyway, let's forget about that correction ;) and return to the issue of this post. So, once more: have you seen in some quite reliable source (I mean say, written by native speaker ;) "Simson's line" really WITH 's?

And what do you think about the following rule (retyped from previous posts):

-------------

It seems that things like "Simson line", or "Green function", or "Prandtl number" are NOT some fixed 'line', 'function', 'number' and they just have the names of Simson, Green and Prandtl. And that's why one shouldn't use posseive case here.

On the other hand "Fermat's theorem", "Zorn's lemma", "Planck's constant" are some concrete "theorem", "lemma" and "constant" and they actually BELONG to the persons who invented them and thus possesive case should be used here.

-------------

-------------------------------

knowledge can become a science

only with a help of mathematics

## Re: second attempt: when to use possesive case???

I did some research about this a long time ago, then I discovered I'm not supposed to write

Pythagoras's theorem but Pythagoras' theorem, but this was already pointed on this thread, otherwise I'm of not much help, but I got a few books behind me and this are some random checks. When nothing else is stated, they were taken from the subject index:

From: Interpolation and Approximation by polynomials (George M. Phillips)

Pappu's theorem

Pascal identities

Euler-Mclaurin formula

Gauss-Legendre rule

Hermite-FÃ©jer operator

Gregory's formula

Kronecker delta

Lagrange interpolation

Minkoski's inequality (ALSO checked on the text, not on index)

Legendre polynomial (checked on the text, not on index)

Chebyshev polynomial (checked on the text, not on index)

Bernoulli numbers (checked on the text, not on index)

From: Concrete Mathematics (Graham, Knuth and Patashnik)

Stirling numbers

Pascal's triangle

Stirling's numbers (as listed on the List o f tables)

Lagrange's identity (checked on the text, not on index)

Wilson's theorem (checked on the text, not on index)

Complex analysis (Lang )

All named theorems in text use 's

but he also uses

Bernoulli polynomial,

Green function

Euler product

Picard-Borel theorem

Dirac family

Poisson family

Poisson kernel

Poisson integral

it seems that when lang refers to a mathematical entity, he doens't uses 's (since mathematicians do not "own" math objects) but when refering to statements, lemmas and theorems, he does use 's

That convention makes sense to me, so I would say

"By the Simson's theorem on Simson lines..."

I'll do more checks soon...

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: second attempt: when to use possesive case???

> it seems that when refering to a mathematical entity,

> [one] doens't uses 's (since mathematicians do not "own" math objects)

> but when refering to statements, lemmas and theorems,

> [one] does use 's

> That convention makes sense to me, so I would say

> "By the Simson's theorem on Simson lines..."

Well, it is something similar to what I said. It would be interesting to find some place where some rule is explicitly stated ;)

> I'll do more checks soon...

I'll also be looking for...

-------------------------------

knowledge can become a science

only with a help of mathematics