A Refinement of Hilbert's 1888 Theorem: Separating Cones along the Veronese Variety
arxiv(2024)
摘要
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.
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