  It is shown that the polynomial $\lambda(t) = {\rm Tr}[(A + tB)^p]$ has
nonnegative coefficients when $p \leq 7$ and A and B are any two complex
positive semidefinite $n \times n$ matrices with arbitrary $n$. This proofs a
general nontrivial case of the Lieb-Seiringer formulation of the
Bessis-Moussa-Villani conjecture which is a long standing problem in
theoretical physics.
