The cone of all positive semidefinite (PSD) real forms in n + 1 (n ≥ 1) variables of degree 2d
(d ≥ 1) contains the subcone of all forms that are representable as finite sums of squares (SOS) of
forms of half degree. In 1888, Hilbert showed in a seminal paper that both cones coincide exactly
in the Hilbert cases n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4).
In this talk, we construct a filtration of intermediate cones between the SOS and PSD cones
along a filtration of varieties containing the Veronese variety and investigate it for proper inclusions
in any non-Hilbert cases. To this end, we establish a sufficient criterion for a given intermediate
cone to coincide with the SOS cone on the one hand. On the other hand, we develop a tool
with which we are able to determine the greatest cone in the filtration containing an a priori
fixed PSD-extremal not-SOS circuit form. This allows us to introduce and prove a refinement of
Hilbert’s 1888 Theorem in three steps. First, we lay out the situation for (n + 1)-ary quartics
(n ≥ 3). Secondly, we extend our findings to (n + 1)-ary sextics (n ≥ 2) and, then, thirdly, to any
non-Hilbert case via a degree-jumping principle.