  Consider the polynomial $tr (A + tB)^m$ in $t$ for positive hermitian
matrices $A$ and $B$ with $m \in \N$. The Bessis-Moussa-Villani conjecture (in
the equivalent form of Lieb and Seiringer) states that this polynomial has
nonnegative coefficients only. We prove that they are at least asymptotically
positive, for the nontrivial case of $AB \neq 0$. More precisely, we show -
once complex-analytically, once combinatorially - that the $k$-th coefficient
is positive for all integer $m \geq m_0$, where $m_0$ depends on $A$, $B$ and
$k$.
