  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.
