Georgia Institute of Technology College of Sciences

A multivariate real polynomial $p$ is nonnegative if $p(x) \geq 0$ for all $x \in R^n$. I will review the history and motivation behind the problem of representing nonnegative polynomials as sums of squares. Such representations are of interest for both theoretical and practical computational reasons. I will present two approaches to studying the differences between nonnegative polynomials and sums of squares. Using techniques from convex geometry we can conclude that if the degree is fixed and the number of variables grows, then asymptotically there are significantly more nonnegative polynomials than sums of squares. For the smallest cases where there exist nonnegative polynomials that are not sums of squares, I will present a complete classification of the differences between these sets based on algebraic geometry techniques.