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 ${\mathrm{\Sigma }}_{n+1,2d}$ of those forms that are representable as finite sums of squares if and only if $(n+1,2d)=(2,2d{)}_{d\ge 1}$ or $(n+1,2{)}_{n\ge 1}$ or $(3,4)$. These are the Hilbert cases. In [GHK23, GHK24], we applied the Gram matrix method to construct cones between ${\mathrm{\Sigma }}_{n+1,2d}$ and ${\mathcal{P}}_{n+1,2d}$, defined by projective varieties containing the Veronese variety. In particular, we introduced and examined a specific cone filtration $${\mathrm{\Sigma }}_{n+1,2d}=C_0\subseteq \dots \subseteq C_n\subseteq C_{n+1}\subseteq \dots \subseteq C_{k(n,d)-n}={\mathcal{P}}_{n+1,2d}$$ 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.