Cyclic Equivalence to Sums of Hermitian Squares

  • Sabine Burgdorf
  • Igor Klep
  • Janez Povh
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


When we move focus from positive semidefinite non-commutative polynomials to trace-positive non-commutative polynomials we naturally meet cyclic equivalence to hermitian squares, see Definitions 1.57 and 1.60 In this chapter we will consider the question whether an nc polynomial is cyclically equivalent to SOHS, i.e., whether it is a member of the cone \(\varTheta ^{2}\), which is a sufficient condition for trace-positivity. A special attention will be given to algorithmic aspects of detecting members in \(\varTheta ^{2}\). We present a tracial version of the Gram matrix method based on the tracial version of the Newton chip method which by using semidefinite programming efficiently answers the question if a given nc polynomial is or is not cyclically equivalent to a sum of hermitian squares.


Matrix Method Positive Semidefinite Semidefinite Programming Quantifier Elimination Positive Semidefinite Matrix 
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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sabine Burgdorf
    • 1
  • Igor Klep
    • 2
  • Janez Povh
    • 3
  1. 1.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  2. 2.Department of MathematicsThe University of AucklandAucklandNew Zealand
  3. 3.Faculty of Information Studies in Novo MestoNovo MestoSlovenia

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