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Sums of Hermitian Squares and the BMV Conjecture

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Abstract

We show that all the coefficients of the polynomial

$$\mathop{\mathrm{tr}}((A+tB)^{m})\in\mathbb{R}[t]$$

are nonnegative whenever m≤13 is a nonnegative integer and A and B are positive semidefinite 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 semidefinite 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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Authors and Affiliations

  1. Oddelek za Matematiko Inštituta za Matematiko, Fiziko in Mehaniko, Univerza v Ljubljani, Jadranska 19, 1111, Ljubljana, Slovenia

    Igor Klep

  2. Laboratoire de Mathématiques, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes cedex, France

    Markus Schweighofer

Authors
  1. Igor Klep
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  2. Markus Schweighofer
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Correspondence to Igor Klep.

Additional information

The first author acknowledges the financial support from the state budget by the Slovenian Research Agency (project No. Z1-9570-0101-06).

Supported by the DFG grant “Barrieren”.

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Klep, I., Schweighofer, M. Sums of Hermitian Squares and the BMV Conjecture. J Stat Phys 133, 739–760 (2008). https://doi.org/10.1007/s10955-008-9632-x

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  • DOI: https://doi.org/10.1007/s10955-008-9632-x

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