Semidefinite and Second Order Cone Programming Seminar Fall 2012 Lecture 10 Instructor : Farid Alizadeh Scribe

1 Overview In this lecture, we show that a number of sets and functions related to the notion of sum-of-squares (SOS) are SD-representable. We will start with positive polynomials. Then, we introduce a general algebraic framework in which the notion of sum-of-squares can be formulated in very general setting. Recall the cone of nonnegative univariate polynomials: P 2d [t] = p(t) = p 0 + p 1 t + p 2 t 2 + · · · + p 2d t 2d 0 ∀t ∈ R Earlier we have examined this cone and have shown that it is SD-representable. We now consider the case of multivariate polynomials. The set of nonnegative polynomials is Recall that in the case of univariate polynomials, a polynomial p(t) 0 for all t ∈ R if and only if there are two polynomials p 1 (t), p 2 (t) such that p(t) = p 2 1 (t) + p 2 2 (t). In other words, a univariate polynomial is nonnegative over the real line if and only if it is a sum of squares. For multivariate polynomials this is no longer true.