Equivalent forms of the Bessis-Moussa-Villani conjecture

Abstract

The BMV conjecture for traces, which states that $Tr exp(A -λB)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $λ\mapsto Tr(A+λB)^p$ has only non-negative coefficients for all $A, B \geq 0$, $p\in N$ and (ii) $λ\mapsto \Tr (A+λB)^{-p}$ is the Laplace transform of a positive measure for $A, B \geq 0$, $p>0$.