Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors

A symmetric positive semi-definite (PSD) tensor, which is not sum-of-squares (SOS), is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? The answer for this question has both theoretical and practical significance. Under the assumptions that the generating vector v of a Hankel tensor A is symmetric and the fifth elementv4of v is fixed at 1, we show that there are two surfacesM0andN0with the elementsv2,v6,v1,v3,v5of v as variables, such thatM0≥N0, A is SOS if and only ifv0≥M0, and A is PSD if and only ifv0≥N0, wherev0is the first element of v. IfM0=N0for a point P=(v2,v6,v1,v3,v5)T, there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for suchv2,v6,v1,v3,v5. Then, we call such P a PNS-free point. We prove that a 45-degree planar closed convex cone, a segment, a ray and an additional point are PNS-free. Numerical tests check various grid points and report that they are all PNS-free.