I just started looking into positive polynomials on compact semi-algebraic sets and it requires a mixture of optimization, functional analysis and real algebraic geometry. I would like to know if you had references to share that discuss this problem or good introductory books or articles on any of the above topics. Research papers are also welcome!
Thanks again!
It is an old question, but let me add some references more oriented toward sums of squares and positive polynomials, rather than general real algebraic geometry, which are also mentioning explicitly the applications in optimization:
The first one is a wonderful book, very clearly written, on sums of squares and positive polynomials. The last chapter discusses the applications to polynomial optimization. The second is a recent book by the leading expert in real algebra. The third one is also recent, and more accessible for an application-oriented audience. The last one is the monograph written by the leading expert in the dual moment problem, mostly from the point of view of functional analysis.
Here are some more introductory resources on semi-algebraic geometry:
Coste's An Introduction to Semialgebraic Geometry: these notes can be used for something between a mini-course and a full course depending on how fast you want to go. I think it provides a generally good introduction, though depending on your specific focus there are some sections you may find aren't quite so necessary.
Coste's An Introduction to O-Minimal Geometry: o-minimal geometry is a generalization of semi-algebraic geometry and you may find need of these concepts. If you're only (for now) interested in the strictly semi-algebraic stuff, this may be more appropriate for skimming or searching rather than reading outright.
Denkowska and Denkowski's A long and winding road to definable sets: this is a historical survey paper which outlines many results and gives a lot of good jumping-off points and quick references.
If you're looking for textbook-length sources, the following are considered pretty solid choices:
Benedetti and Risler's Real algebraic and semi-algebraic sets: This starts off fairly similar in scope to Coste's Introduction, but it's a bit longer and gets more in to real algebraic geometry.
Bochnak, Coste, and Roy's Real Algebraic Geometry: I've only used this as a reference work, but it does a good job at that.
