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

For n , d ∈ ℕ, the cone 𝓟 n +1,2 d of positive semidefinite real forms in n + 1 variables of degree 2 d contains the subcone Σ n +1,2 d 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, 2 d ) with n + 1 = 2 or 2 d = 2 or ( n + 1, 2 d ) = (3, 4). In this paper, we induce a filtration of intermediate cones between Σ n +1,2 d and 𝓟 n +1,2 d via the Gram matrix approach in [4] on a filtration of irreducible projective varieties V k − n ⊊ … ⊊ V n ⊊ … ⊊ 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 +1,2 d . 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.