Lectures on nonnegative polynomials and sums of squares

Abstract

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic geometry and commutative algebra. We briefly discuss the connection to semidefinite programming and applications. We expect the reader to be familiar with algebra and the fundamentals of algebraic geometry, on the level of a graduate course, and know basic facts about positive semidefinite matrices. For additional background, we refer to Eisenbud (1995) for commutative algebra, Barvinok (2002) for convex geometry, and Blekherman, Parrilo, and Thomas (2013) for polynomial optimization.