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

For n,d∈ℕ, the cone 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 Σ_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 Σ_n+1,2d=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 Σ_n+1,2d=C_0⊆…⊆ C_n ⊆ C_n+1⊆…⊆ C_k(n,d)-n=𝒫_n+1,2d 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 μ(n,d) of strictly separating intermediate cones (i.e., C_i such that Σ_n+1,2d⊊ C_i ⊊𝒫_n+1,2d) for the cases (3,6) and (n+1,2d)_n≥ 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 μ(n,d). The above cone filtration thus reduces to a specific cone subfiltration Σ_n+1,2d=C_0^'⊊ C_1^'⊊…⊊ C_μ(n,d)^'⊊ C_μ(n,d)+1^'=𝒫_n+1,2d in which each inclusion is strict. Secondly, we show that each C_i^', and hence each strictly separating C_i, fails to be a spectrahedral shadow.