2024/01/08 by Charu Goel, Goel, Charu, Sarah Hess +3
Mathematics · Computer Science · #Tensor decomposition and applications #Polynomial and algebraic computation #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2401.03813
For n,d∈ℕ, the cone Pn+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 Σn+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 Σn+1,2d=Pn+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 Σn+1,2d=C0⊆ … ⊆ Cn ⊆ Cn+1 ⊆ … ⊆ Ck(n,d)-n=Pn+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., Ci such that Σn+1,2d\subsetneq Ci \subsetneq Pn+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 Σn+1,2d=C0^′\subsetneq C1^′ \subsetneq … \subsetneq Cμ(n,d)^′ \subsetneq Cμ(n,d)+1^′=Pn+1,2d in which each inclusion is strict. Secondly, we show that each Ci^′, and hence each strictly separating Ci, fails to be a spectrahedral shadow.