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.