Nonnegative Polynomials and Sums of Squares

A polynomial with real coefficients is nonnegative if it takes only nonnegative values. For example, any sum of squares is obviously nonnegative. Sums of squares are also important from an algorithmic standpoint, since testing a polynomial is a sum of squares is a semidefinite programming problem. After reviewing the history of the problem I will present an approach to studying the relationship between polynomials nonnegative on a variety and sums of squares based on convex algebraic geometry. This will result in the classification of the cases of equality, and degree bounds for rational sums of squares representations on curves. The talk is based on joint work with Sadik Iliman, Martina Kubitzke, Greg Smith and Mauricio Velasco.