  It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive
coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian
positive definite matrices. This case is the first that is not covered by
prior, general results. This problem arises from a conjecture raised by Bessis,
Moussa and Villani in connection with a long-standing problem in theoretical
physics. The full conjecture, as shown recently by Lieb and Seiringer, is
equivalent to $p(t)$ having positive coefficients for any $m$ and any two
$n$-by-$n$ positive definite matrices. We show that, generally, the question in
the real case reduces to that of singular $A$ and $B$, and this is a key part
of our proof.
