Let respectively denote the cone of polynomials in
real unknowns of degree that are positive respectively that can be written as sums of squares of polynomials. The difference of these two cones is important in
the theory of sums of squares approaches for polynomial optimization.
In this talk I will discuss this question in
the setting of symmetric polynomials and discuss the relation of the
asymptotical behaviour of these cones to symmetric inequalities.
In this context I will proove that in contrast to the non-symmetric
case as tends to infinity there are not substantially more
positive polynomials then sums of squares. In particular in the case
of symmetric forms of degree 4 the cones coincide asymptotically