## Sums of hermitian squares and the BMV conjecture

2008
Journal article
##### Published in
Journal of Statistical Physics ; 133 (2008), 4. - pp. 739-760. - ISSN 0022-4715
##### Abstract
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.
510 Mathematics
##### Keywords
Bessis-Moussa-Villani (BMV) conjecture,sum of hermitian squares,trace inequality,semide nite programming.
##### Cite This
ISO 690KLEP, 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
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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<dcterms:abstract xml:lang="eng">Abstract. We show that all the coe cients of the polynomial tr((A + tB)&lt;sup&gt;m&lt;/sup&gt;)	&amp;#8712; ℝ[t] are nonnegative whenever m &amp;#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 &amp;#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.</dcterms:abstract>
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Yes