The even Vandermonde map

In this talk we consider the geometry of the image of the even Vandermonde map $\mathbb{R}^n$ to $\mathbb{R}^d$ consisting of the first d even power sums. For fixed degree d, the images form an increasing chain in the number of variables. We give a description of the image for finite n and at infinity, and prove that the image has the combinatorial structure of a cyclic polytope.

We show how the image of the Vandermonde map relates to the study of copositive symmetric forms and prove undecidability of verifying nonnegativity of trace polynomials whose domains are all symmetric matrices of all sizes.

This is joint work with Jose Acevedo, Greg Blekherman and Cordian Riener.