A polynomial of n variables with real coefficients is said to be positive semi-definite if it is non-negative for all variables with arbitrary real values. Hilbert's 17th problem asks whether one can express a positive semi-definite polynomial as a sum of squares of rational functions.

I will briefly review the classical results, then I'll talk about a recent progress of Benoist in low degree. The main ingredients are geometric interpretation of Hilbert's 17th problem, Bloch-Ogus theory, Voevodsky's proof of Milnor conjecture and cohomological computation.