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Deciding non-negativity of a polynomial in n variables with real coefficients in the n-dimensional real space is an important problem that appears in optimization and several other areas. We know that if a polynomial can be written as a sum of squares of polynomials, it takes only non-negative values. However, the converse doesn't always hold. On the other hand, Artin in 1927 proved that the non-negativity implies being a sum-of-squares of rational functions, which also completed the answer of Hilbert's 17th problem. Today, sums-of-squares play an important role in real algebraic geometry and polynomial optimization.

In this talk, we introduce some concepts from real algebraic geometry such as sums-of-squares, hyperbolic polynomials, spectrahedral representability,  the generalized Lax conjecture, and we will talk about the relation between them in the context of non-negativity.