A Refinement of Hilbert's 1888 Theorem: Separating Cones along the Veronese Variety

For $n,d\in\mathbb{N}$, the cone $\mathcal{P}_{n+1,2d}$ of positive semi-definite (PSD) $(n+1)$-ary $2d$-ic forms (i.e., homogeneous polynomials with real coefficients in $n+1$ variables of degree $2d$) contains the cone $\Sigma_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of $(n+1)$-ary $d$-ic forms. Hilbert's 1888 Theorem states that $\Sigma_{n+1,2d}=\mathcal{P}_{n+1,2d}$ exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(3,4)$. For the non-Hilbert cases, we examine in [GHK] a specific cone filtration \begin{equation} \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}\end{equation} along $k(n,d)+1-n$ projective varieties containing the Veronese variety via the Gram matrix method. Here, $k(n,d)+1$ is the dimension of the real vector space of $(n+1)$-ary $d$-ic forms. In particular, we compute the number $\mu(n,d)$ of strictly separating intermediate cones (i.e., $C_i$ such that $\Sigma_{n+1,2d}\subsetneq C_i \subsetneq \mathcal{P}_{n+1,2d}$) for the cases $(3,6)$ and $(n+1,2d)_{n\geq 3,d=2,3}$. In this paper, firstly, we generalize our findings from [GHK] to any non-Hilbert case by identifying each strict inclusion in the above cone filtration. This allows us to give a refinement of Hilbert's 1888 Theorem by computing $\mu(n,d)$. The above cone filtration thus reduces to a specific cone subfiltration \begin{equation} \Sigma_{n+1,2d}=C_0^\prime\subsetneq C_1^\prime \subsetneq \ldots \subsetneq C_{\mu(n,d)}^\prime \subsetneq C_{\mu(n,d)+1}^\prime=\mathcal{P}_{n+1,2d} \end{equation} in which each inclusion is strict. Secondly, we show that each $C_i^\prime$, and hence each strictly separating $C_i$, fails to be a spectrahedral shadow.