The University of Michigan Combinatorics Seminar


Abstract 

We study metric properties of the cone of homogeneous
nonnegative multivariate polynomials and the cone of sums of
powers of linear forms, and the relationship between the two
cones. We compute the maximum volume ellipsoid of the natural base
of the cone of nonnegative polynomials and the minimum volume
ellipsoid of the natural base of the cone of powers of linear
forms and compute the coefficients of symmetry of the bases. The
multiplication by
(x_{1}^{2}+...+x_{n}^{2})^{m}
induces an
isometric embedding of the space of polynomials of degree 2k
into the space of polynomials of degree 2(k+m), which allows us
to compare the cone of nonnegative polynomials of degree 2k and
the cone of sums of 2(k+m)powers of linear forms. We estimate
the volume ratio of the bases of the two cones and the rate at
which it approaches 1 as m grows.
