Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms

An important theorem by Timofte states that nonnegativity of real $n$-variate symmetric polynomials of degree $d$ can be decided at test sets given by all points with at most $\lfloor\frac{d}{2}\rfloor$ distinct components. However, if the degree is sufficiently larger than the number of variables, then the theorem obviously does not provide nontrivial information. Our approach is to look at $(m + 1)$-dimensional subspaces of even symmetric forms of degree 4d, at which nonnegativity can be checked at $(m - 1)$-points, i.e., points with at most $m - 1 \in \N$ distinct components, where $m$ is independent of the degree of the forms and better than Timofte's bound. Furthermore, for fixed $k \in \N$, we tackle problems concerning the maximum dimension of such subspaces, at which nonnegativity can be checked at all $k$-points, as well as the geometrical and topological structure of the set of all forms whose nonnegativity can be decided at all $k$-points.