Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

  • Sabine Burgdorf
  • Kristijan Cafuta
  • Igor Klep
  • Janez Povh


This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of Hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several examples are given illustrating the results.


Sum of squares Semidefinite programming Noncommutative polynomial Matlab toolbox Newton polytope Free positivity 



The authors thank the three anonymous referees whose attentive comments helped improve the paper.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Sabine Burgdorf
    • 1
  • Kristijan Cafuta
    • 2
  • Igor Klep
    • 3
  • Janez Povh
    • 4
  1. 1.SB-MATHGEOM-EGGEPFLLausanneSwitzerland
  2. 2.Fakulteta za elektrotehniko, Laboratorij za uporabno matematikoUniverza v LjubljaniLjubljanaSlovenia
  3. 3.Department of MathematicsThe University of AucklandAucklandNew Zealand
  4. 4.Fakulteta za informacijske študije v Novem mestuNovo mestoSlovenia

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