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Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

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Abstract

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.

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Fig. 1
Algorithm 1
Algorithm 2

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Notes

  1. A BMV polynomial S m,k (X,Y) is the sum of all words in X,Y of total degree m and degree k in Y.

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Acknowledgements

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

Author information

Authors and Affiliations

  1. SB-MATHGEOM-EGG, EPFL, 1015, Lausanne, Switzerland

    Sabine Burgdorf

  2. Fakulteta za elektrotehniko, Laboratorij za uporabno matematiko, Univerza v Ljubljani, Tržaška 25, 1000, Ljubljana, Slovenia

    Kristijan Cafuta

  3. Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Igor Klep

  4. Fakulteta za informacijske študije v Novem mestu, Novi trg 5, 8000, Novo mesto, Slovenia

    Janez Povh

Authors
  1. Sabine Burgdorf
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  2. Kristijan Cafuta
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  3. Igor Klep
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  4. Janez Povh
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Corresponding author

Correspondence to Janez Povh.

Additional information

S. Burgdorf was partially supported by the Zukunftskolleg Konstanz. I. Klep was supported by the Faculty Research Development Fund (FRDF) of The University of Auckland (project no. 3701119). Partially supported by the Slovenian Research Agency (program no. P1-0222). Part of this research was done while the author was on leave from the University of Maribor. S. Burgdorf and I. Klep were partially supported by the French-Slovene partnership project Proteus 20208ZM. J. Povh was supported by the Slovenian Research Agency (program no. P1-0297).

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Burgdorf, S., Cafuta, K., Klep, I. et al. Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials. Comput Optim Appl 55, 137–153 (2013). https://doi.org/10.1007/s10589-012-9513-8

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