Interesting question. You might want to have a look at the following books...

 - Marshall: Positive polynomials and sums of squares
 - Prestel: Positive polynomials
 - Bochnak, Coste, Roy: Real algebraic geometry
 - Basu, Pollack, Roy: Algorithms in real algebraic geometry
 - Knebusch, Scheiderer: Einführung in die reelle Algebra
 - Andradas, Bröcker, Ruiz: Constructible sets in real geometry

...and the following articles...

- http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf
- http://www.math.uni-konstanz.de/~schweigh/publications/sosdualsdp.pdf

Also the so-called "S-procedure" could be of interest for you.