Speaker: Greg Blekherman

Title: There are Significantly More Nonnegative Polynomials Than Sums of Squares

Abstract:

It is a classical result due to Hilbert that, with few exceptions, for a fixed degree and number of variables, there exist nonnegative polynomials that are not sums of squares of polynomials. The talk focus on the quantitative relationship between the cone of nonnegative polynomials and the cone of sums of squares of polynomials. I will discuss how to derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the two cones. The bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two it follows that there are significantly more non-negative polynomials than sums of squares. Moreover, the asymptotical discrepancy between the cones grows as the degree increases. The above results show that it is in general not suitable to replace testing for non-negativity with testing for sums of squares.
Time permitting, I will explain a unified framework for the methods of the proof, which deals with the geometry of convex hulls of orbits of points under a group action, and discuss applications to other objects of the same form, such as the Traveling Salesman Polytope.