Equivalent forms of the Bessis-Moussa-Villani conjecture

TL;DR

This paper demonstrates that the BMV conjecture for traces is equivalent to two other positivity conditions involving polynomial coefficients and Laplace transforms of inverse matrices, providing new perspectives on the conjecture.

Contribution

It establishes the equivalence of the BMV conjecture with two alternative statements, offering new approaches to its proof.

Findings

Proves the equivalence of the BMV conjecture to polynomial coefficient positivity.

Shows the Laplace transform representation of inverse matrix traces.

Provides new insights into the structure of the BMV conjecture.

Abstract

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