Hilbert's seventeenth problem

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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.

Algorithm was found by Delzell, see his article "A continuous, constructive solution to Hilbert's 17th problem."

The nature of the question is because there are polynomials (example: f(x,y,z) = z⁶ + x⁴z² + x²y⁴ − 3x²y²z²) which are non-negative over reals and 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 ([1]). However every real nonnegative polynomial f can be approximated as closely as desired (in the l₁-norm of its coefficient vector) by a sequence of polynomials {f_ε} that are sums of squares of polynomials [2].

It's 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 (2008) is , for details see:

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

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