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 AA is symmetric and the fifth element v4v4 of v is fixed at 11, we show that there are two surfaces M0M0 and N0N0 with the elements v2,v6,v1,v3,v5v2,v6,v1,v3,v5 of v as variables, such that M0≥N0M0≥N0, AA is SOS if and only if v0≥M0v0≥M0, and AA is PSD if and only if v0≥N0v0≥N0, where v0v0 is the first element of v. If M0=N0M0=N0 for a point P=(v2,v6,v1,v3,v5)⊤P=(v2,v6,v1,v3,v5)⊤, there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such v2,v6,v1,v3,v5v2,v6,v1,v3,v5. Then, we call such PP a PNS-free point. We prove that a 4545-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.