Sums of hermitian squares and the BMV conjecture

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2008
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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.

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Fachgebiet (DDC)
510 Mathematik
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Bessis-Moussa-Villani (BMV) conjecture, sum of hermitian squares, trace inequality, semide nite programming.
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ISO 690KLEP, Igor, Markus SCHWEIGHOFER, 2008. Sums of hermitian squares and the BMV conjecture. In: Journal of Statistical Physics. 2008, 133(4), pp. 739-760. ISSN 0022-4715. Available under: doi: 10.1007/s10955-008-9632-x
BibTex
@article{Klep2008hermi-15620,
  year={2008},
  doi={10.1007/s10955-008-9632-x},
  title={Sums of hermitian squares and the BMV conjecture},
  number={4},
  volume={133},
  issn={0022-4715},
  journal={Journal of Statistical Physics},
  pages={739--760},
  author={Klep, Igor and Schweighofer, Markus}
}
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