Many difficult optimization problems can be modeled as the optimization of a polynomial over a semi-algebraic set. This last problem has seen enormous progress over the last decades thanks to the progress in the theory of nonnegative polynomials (Positivstellensätze by Putinar, Schmüdgen, etc.) and the discovery of hierarchies of sums of squares (by Lasserre and Parrilo). A recruited PhD student will be trained in the field of real algebraic geometry under the supervision of Khazhgali Kozhasov (Chaire de Professeur Junior). One of the objectives is to enrich the state-of-the-art knowledge on the geometry of nonnegative polynomials (in particular sums of squares). ​The results obtained in the thesis will be applied to the study of the structural properties of polynomial optimization problems.