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

Yannan Chen, Liqun Qi, Qun Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

12 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)356-368
Number of pages13
JournalJournal of Computational and Applied Mathematics
Publication statusPublished - 15 Aug 2016


  • Generating vector
  • Hankel tensor
  • PNS-free
  • Positive semi-definiteness
  • Sum of squares

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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