The BMV conjecture in algebraic form

Let $A$ and $B$ be $n\times n$ positive semidefinite matrices, and let $m$ be a nonnegative integer. Define

$$p:=\operatorname{tr}((A+tB)^m)\in\mathbb{R}[t].$$

BMV conjecture, algebraic form. The polynomial $p$ has only nonnegative coefficients. This is an algebraic reformulation of the Bessis–Moussa–Villani conjecture, which originated in quantum statistical mechanics and is equivalent to complete monotonicity of $t\mapsto\operatorname{tr}(e^{A-tB})$. The source describes the conjecture as unresolved at this point, with extensive partial results and approaches toward a proof.