Elementary root separation arguments imply that a non-negative polynomial on the real line is a sum of two squares of polynomials. A non-constructive and ingenious enumerative geometry observation of Hilbert shows that in two or more variables, non-negative polynomials are not always sums of squares. This led to Hilbert’s 17-th problem, asking whether such a decomposition is possible in the field of rational functions. The answer is yes, due to Emil Artin. The gap between polynomials which are positive on semi-algebraic sets and corresponding weighted sums of squares was elucidated by Tarski’s elimination of quantifiers principle. The first part of the lecture will contain accessible details and historical notes on these topics, now part of Real Algebra and Real Algebraic Geometry.

In the second part of the lecture, I will show how sums of squares decompositions led F. Riesz to the definitive form of the spectral theorem for self-adjoint transforms of a Hilbert space. Implying for instance novel positivity results of harmonic analysis. Reversing the historical arrow, I will show how operator theory has put some essential marks on purely Real Algebra chapters. We will also touch positivity in non-commutative *-algebras and Lie-algebras.

The third part of the lecture will contain applications of relatively recent positivity certificates to global, non-convex optimization, stability of dynamical systems and construction of wavelet frames.

The video of this talk is available on the IISc Math Department channel.