Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

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 $k$.