Sums of Hermitian squares as an approach to the BMV conjecture

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani conjecture that all coefficients of the polynomial p(t) = tr((A +tr B)m) are non-negative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m the coefficient of t4 in p(t) is non-negative, using a connection to sums of Hermitian squares of non-commutative polynomials which has been established by Klep and Schweighofer. This implies by a well-known result of Hillar that the coefficients of tk are non-negative for 0 k 4, and by symmetry as well for m epsilon k epsilon m - 4.