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An important theorem by Timofte states that nonnegativity of real $n$-variatesymmetric polynomials of degree $d$ can be decided at test sets given by allpoints with at most $\lfloor\frac{d}{2}\rfloor$ distinct components. However,if the degree is sufficiently larger than the number of variables, then thetheorem obviously does not provide nontrivial information. Our approach is tolook 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 atmost $m - 1 \in \N$ distinct components, where $m$ is independent of the degreeof the forms and better than Timofte's bound. Furthermore, for fixed $k \in\N$, we tackle problems concerning the maximum dimension of such subspaces, atwhich nonnegativity can be checked at all $k$-points, as well as thegeometrical and topological structure of the set of all forms whosenonnegativity can be decided at all $k$-points.

Low Dimensional Test Sets for Nonnegativity of Even Symmetric Forms