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# Sums of hermitian squares and the BMV conjecture

Type of Publication: | Journal article |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-156200 |

Author: | Klep, Igor; Schweighofer, Markus |

Year of publication: | 2008 |

Published in: | Journal of Statistical Physics ; 133 (2008), 4. - pp. 739-760. - ISSN 0022-4715 |

DOI (citable link): | https://dx.doi.org/10.1007/s10955-008-9632-x |

Summary: |
Abstract. We show that all the coe cients of the polynomial tr((A + tB)
^{m}) ∈ ℝ[t] are nonnegative whenever m ≤ 13 is a nonnegative integer and A and B are positive semide nite matrices of the same size. This has previously been known only for m ≤ 7. The validity of the statement for arbitrary m has recently been shown to be equivalent to the Bessis-Moussa-Villani conjecture from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semide nite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators. |

Subject (DDC): | 510 Mathematics |

Keywords: | Bessis-Moussa-Villani (BMV) conjecture, sum of hermitian squares, trace inequality, semide nite programming. |

Link to License: | Terms of use |

Bibliography of Konstanz: | Yes |

Checksum:
MD5:8434214e68a9af772dee4bca634d545d

KLEP, Igor, Markus SCHWEIGHOFER, 2008. Sums of hermitian squares and the BMV conjecture. In: Journal of Statistical Physics. 133(4), pp. 739-760. ISSN 0022-4715. Available under: doi: 10.1007/s10955-008-9632-x

@article{Klep2008hermi-15620, title={Sums of hermitian squares and the BMV conjecture}, year={2008}, doi={10.1007/s10955-008-9632-x}, number={4}, volume={133}, issn={0022-4715}, journal={Journal of Statistical Physics}, pages={739--760}, author={Klep, Igor and Schweighofer, Markus} }

bmv.pdf | 102 |