Equivalent Forms of the Bessis–Moussa–Villani Conjecture

Abstract

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