theorems on sums of squares
A generalization of the above is the following:
Theorem (Pfister’s Theorem).
Let be a field of characteristic not . The sum of squares identity of the form
where each is a rational function of and (element of ), is possible iff is a power of .
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 is a field and is a power of 2, then there exists a sum of squares identity of the form
such that each is a rational function of the and a linear function of the , or that
Conversely, if is not a power of , then there exists a field such that the above sum of square identity does not hold for any . Notice that is no longer required to be a linear function of the anymore.
When is the field of reals , we have the following generalization, also due to Pfister:
If is positive semidefinite, then can be written as a sum of 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 indeterminates over the reals, as a sum of squares of rational functions in 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.
- 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|
|Date of creation||2013-03-22 15:06:04|
|Last modified on||2013-03-22 15:06:04|
|Last modified by||CWoo (3771)|
|Defines||Hilbert’s 17th Problem|