Sums of hermitian squares and the BMV conjecture

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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} }

Klep, Igor eng 2011-10-31T09:23:01Z Schweighofer, Markus deposit-license 2011-10-31T09:23:01Z Klep, Igor Sums of hermitian squares and the BMV conjecture Abstract. We show that all the coe cients of the polynomial tr((A + tB)<sup>m</sup>) &#8712; ℝ[t] are nonnegative whenever m &#8804; 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 &#8804; 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. Schweighofer, Markus First publ. in: Journal of Statistical Physics 133 (2008), 4. - S. 739-760 2008

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