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Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors. (English) Zbl 1334.15025

Summary: 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 \(\mathbf{v}\) of a Hankel tensor \(\mathcal{A}\) is symmetric and the fifth element \(v_4\) of \(\mathbf{v}\) 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 \(\mathbf{v}\) as variables, such that \(M_0 \geq N_0\), \(\mathcal{A}\) is SOS if and only if \(v_0 \geq M_0\), and \(\mathcal{A}\) is PSD if and only if \(v_0 \geq N_0\), where \(v_0\) is the first element of \(\mathbf{v}\). If \(M_0 = N_0\) for a point \(P = (v_2, v_6, v_1, v_3, v_5)^\top\), 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.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A69 Multilinear algebra, tensor calculus
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