Positive Semi-Definiteness and Sum-of-Squares Property of Fourth Order Four Dimensional Hankel Tensors

Abstract

A positive semi-definite (PSD) tensor which is not a sum-of-squares (SOS) tensor is called a PSD non-SOS (PNS) tensor. Is there a fourth order four dimensional PNS Hankel tensor? Until now, this question is still an open problem. Its answer has both theoretical and practical meanings. We assume that the generating vector $v$ of the Hankel tensor $A$ is symmetric. Under this assumption, we may fix the fifth element $v_4$ of $v$ 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 $M_0\ge N_0$, $A$ is SOS if and only if $v_0\ge M_0$, and $A$ is PSD if and only if $v_0\ge N_0$, where $v_0$ is the first element of $v$. If $M_0=N_0$ for a point $P=(v_2,v_6,v_1,v_3,v_5{)}^{\top }$, then 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 a point $P$ PNS-free. We show 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 find that they are also PNS-free.

References in corpus (2)

• SOS-Hankel Tensors: Theory and Application
• Positive Semi-Definiteness of Generalized Anti-Circulant Tensors

Cited by in corpus (1)

• Three Dimensional Strongly Symmetric Circulant Tensors