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 element v 4 of v is fixed at 1 , we show that there are two surfaces M 0 and N 0 with the elements v 2 , v 6 , v 1 , v 3 , v 5 of v as variables, such that M 0 ? N 0 , A is SOS if and only if v 0 ? M 0 , and A is PSD if and only if v 0 ? N 0 , where v 0 is the first element of v . If M 0 = N 0 for a point P = ( v 2 , v 6 , v 1 , v 3 , v 5 ) ? , there are no fourth order four dimensional PNS Hankel tensors with symmetric generating vectors for such v 2 , v 6 , v 1 , v 3 , v 5 . 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.

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