The most appealing statement of the Bessis-Moussa-Villani conjecture is as follows:



> 
> **Conjecture:** For all Hermitian positive semidefinite $n\times n$ matrices $A$ and $B$,
>  and all positive integer $m$, the polynomial function
>  $$t \in \mathbb{R}\mapsto g(t) \equiv tr[(A + t B)^m] =
> \sum\limits\_{
> k=0}^m
> a\_kt^k$$
>  has only nonnegative coefficients $a\_k, k=1,\cdots,m$.
> 
> 
> 


Most recent and past research concerns on the quantities $m$ and $n$. What about trying to prove the conjecture in the following way: first show that $a\_m\ge0$, $a\_0\ge0$, $a\_{m-1}\ge0$, $a\_1\ge0$, $a\_{m-2}\ge0$, $a\_2\ge0$. And then go on to show $a\_3\ge 0$ (and so is $a\_{m-3}\ge 0$)...


But it seems difficult to show $a\_3\ge 0$. Can anyone share some idea on this particular coefficient?


**UPDATED**
What is the largest term in $S\_{2m,m}(AB)$ ? More precisely, consider the word in two positive definite letters $A^{k\_1}BA^{k\_2}B\cdots A^{k\_m}B$ , where $(k1,k2,\cdots,k\_m)$ is a pair of nonnegative integer solution of $k\_1+k\_2+\cdots+k\_m=m$ . Is it true that $tr(A^{k\_1}BA^{k\_2}B\cdots A^{k\_m}B)\le tr(A^mB^m)$ ?



What's known about the 3rd coefficient in the BMV conjecture?
Look at <http://arxiv.org/abs/0802.1153>.


Here the fourth coefficient is shown to be non-negative which implies by 
Hillar's decent theorem 
<http://arxiv.org/abs/math/0507166>
also the third coefficient to be non-negative. 


