Sums of Hermitian Squares and the Bmv Conjecture

We show that all the coefficients of the polynomial tr((A+ tB)) ∈ 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.