2024/11/10 by Charu Goel, Goel, Charu, Sarah Hess +3
Computer Science · #11E25 #11E76 #14A10 #14P10 #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Search Problems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2411.06468
openalex publication_date 2024/11/10 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
In 1888, Hilbert proved that the cone Pn+1,2d of positive semidefinite forms in n+1 variables of degree 2d coincides with its subcone Σn+1,2d of those forms that are representable as finite sums of squares if and only if (n+1,2d) = (2,2d)d≥1 or (n+1,2)n≥1 or (3,4). These are the Hilbert cases. In [GHK23, GHK24], we applied the Gram matrix method to construct cones between Σn+1,2d and Pn+1,2d, defined by projective varieties containing the Veronese variety. In particular, we introduced and examined a specific cone filtration Σn+1,2d = C0 ⊆ … ⊆ Cn ⊆ Cn+1 ⊆ … ⊆ Ck(n,d)-n = Pn+1,2d and determined each strict inclusion in non-Hilbert cases. This gave us a refinement of Hilbert's 1888 theorem. Here, k(n,d)+1 is the dimension of the vector space of forms in n+1 variables of degree d. In this paper, we show that the intermediate cones Ci's are closed and describe their interiors and boundaries. We discuss the membership problem for the Ci's, present open problems concerning their dual cones and generalizations to cones defined by toric varieties.