Cones between the cones of positive semidefinite forms and sums of squares

Abstract

For n, d ∈ ℕ, the cone 𝓟 n+1,2d of positive semidefinite real forms in n + 1 variables of degree 2d contains the subcone Σ n+1,2d of those representable as finite sums of squares of real forms. Hilbert [11] 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). In this paper, we induce a filtration of intermediate cones between Σ n+1,2d and 𝓟 n+1,2d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k−n ⊊ … ⊊ Vn ⊊ … ⊊ V 0 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 V 0, …, V n (and V n+1 when n = 2) are varieties of minimal degree, we demonstrate that the corresponding intermediate cones coincide with Σ n+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.