Sums of Hermitian Squares and the BMV Conjecture

We show that all the coefficients of the polynomial 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. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorized users. 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”.