Publikation: Sums of hermitian squares and the BMV conjecture
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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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KLEP, 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-xBibTex
@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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