2023/03/23 by Charu Goel, Goel, Charu, Sarah Hess +3
Engineering · Mathematics · #11E25 #11E76 #14A10 #14P10 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Tensor decomposition and applications #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2303.13178
openalex publication_date 2023/03/23 · openalex created_date 2023/03/25 · openalex updated_date 2026/07/28
The cone Pn+1,2d (n,d∈ℕ) of all positive semidefinite (PSD) real forms in n+1 variables of degree 2d contains the subcone Σn+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 Σn+1,2d\subsetneqPn+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 Σn+1,2d and Pn+1,2d. This filtration is induced via the Gram matrix approach (by Choi, Lam and Reznick) on a filtration of irreducible projective varieties Vk-n\subsetneq … \subsetneq Vn \subsetneq … \subsetneq V0 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 V0,…,Vn are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σn+1,2d. Likewise, for the special case when n=2, Vn+1 is also a variety of minimal degree and the corresponding intermediate cone also coincides with Σn+1,2d. We moreover prove that, in the non-Hilbert cases of (n+1)-ary quartics for n≥ 3 and (n+1)-ary sextics for n≥ 2, all the remaining cone inclusions are strict.