Intermediate Cones between the Cones of Positive Semidefinite Forms and Sums of Squares

The cone $\mathcal{P}_{n+1,2d}$ ($n,d\in\mathbb{N}$) of all positive semidefinite (PSD) real forms in $n+1$ variables of degree $2d$ contains the subcone $\Sigma_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of real forms of half degree $d$. In 1888, Hilbert proved that these cones coincide exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(n+1,2d)=(3,4)$. To establish the strict inclusion $\Sigma_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ in any non-Hilbert case, one can show that verifying the assertion in the basic non-Hilbert cases $(4,4)$ and $(3,6)$ suffices. In this paper, we construct a filtration of intermediate cones between $\Sigma_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$. This filtration is induced via the Gram matrix approach (by Choi, Lam and Reznick) on a filtration of irreducible projective varieties $V_{k-n}\subsetneq \ldots \subsetneq V_n \subsetneq \ldots \subsetneq V_0$ containing the Veronese variety. Here, $k$ is the dimension of the vector space of real forms in $n+1$ variables of degree $d$. By showing that $V_0,\ldots,V_n$ are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with $\Sigma_{n+1,2d}$. Likewise, for the special case when $n=2$, $V_{n+1}$ is also a variety of minimal degree and the corresponding intermediate cone also coincides with $\Sigma_{n+1,2d}$. We moreover prove that, in the non-Hilbert cases of $(n+1)$-ary quartics for $n\geq 3$ and $(n+1)$-ary sextics for $n\geq 2$, all the remaining cone inclusions are strict.