Separating Cones defined by Toric Varieties: Some Properties and Open Problems

In 1888, Hilbert proved that the cone $\mathcal{P}_{n+1,2d}$ of positive semidefinite forms in $n+1$ variables of degree $2d$ coincides with its subcone $\Sigma_{n+1,2d}$ of those forms that are representable as finite sums of squares if and only if $(n+1,2d) = (2,2d)_{d\geq1}$ or $(n+1,2)_{n\geq1}$ or $(3,4)$. These are the Hilbert cases. In [GHK23, GHK24], we applied the Gram matrix method to construct cones between $\Sigma_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$, defined by projective varieties containing the Veronese variety. In particular, we introduced and examined a specific cone filtration $$\Sigma_{n+1,2d} = C_0 \subseteq \ldots \subseteq C_n \subseteq C_{n+1} \subseteq \ldots \subseteq C_{k(n,d)-n} = \mathcal{P}_{n+1,2d}$$ and determined each strict inclusion in non-Hilbert cases. This gave us a refinement of Hilbert's 1888 theorem. Here, $k(n,d)+1$ is the dimension of the vector space of forms in $n+1$ variables of degree $d$. In this paper, we show that the intermediate cones $C_i$'s are closed and describe their interiors and boundaries. We discuss the membership problem for the $C_i$'s, present open problems concerning their dual cones and generalizations to cones defined by toric varieties.