Hilbert's seventeenth problem

Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails expression of definite rational functions as quotients of sums of squares. Original Hilbert's question was:

Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?

This was solved in the affirmative, in 1927, by Emil Artin.

An algorithm was later found by Delzell: see his article [http://www.springerlink.com/content/m18152477392j1t2/ "A continuous, constructive solution to Hilbert's 17th problem."]

A generalization to the matrix case (matrices with rational function entries that are always positive semidefinite are sums of symmetric squares) was given by Gondard, Ribenboim [http://www.ams.org/mathscinet-getitem?mr=432613] and Procesi, Schacher [http://www.ams.org/mathscinet-getitem?mr=432612] , with an elementary proof given by Hillar and Nie [http://arxiv.org/abs/math/0610388] .

The formulation of the question takes into account that there are polynomials, for example

:$f(x,y,z)=z^6+x^4z^2+x^2y^4-3x^2y^2z^2$

which are non-negative over reals and yet which cannot be represented as a sum of squares of other polynomials. This example was taken from:

Marie-Francoise Roy. The role of Hilbert’s problems in real algebraic geometry.Proceedings of the ninth EWM Meeting, Loccum, Germany 1999.

Explicit sufficient conditions for a polynomial f to be a sum of squares of other polynomials were found ( [http://www.optimization-online.org/DB_HTML/2007/02/1587.html] , [http://www.mathcs.emory.edu/~vicki/pub/sos.pdf] ). However every real nonnegative polynomial f can be approximated as closely as desired (in the $l\_1$-norm of its coefficient vector) by a sequence of polynomials $\{f\_epsilon\}$ that are sums of squares of polynomials [http://portal.acm.org/citation.cfm?id=1330215.1330223&coll=GUIDE&dl=] .

It is an open question what is the smallest number

:$v(n,d)$,

such that any n-variate, non-negative polynomial of degree d can be written as sum of at most

:$v(n,d)$

rational functions over the reals.

The best known result (as of 2008) is

:$v(n,d)leq2^n$;

for details see:

A. Pfister. Zur Darstellung definiter Funktionen als Summe von Quadraten. Invent.Math. 4 (1967), 229–237