# Stahl's Theorem (aka BMV Conjecture): Insights and Intuition on its Proof

## TL;DR

This paper provides a concise, self-contained presentation of Stahl's proof of the BMV conjecture, which states that the trace of the exponential of certain matrices is a Laplace transform of a positive measure.

## Contribution

It offers a streamlined, self-contained version of Stahl's proof, making the complex argument more accessible and easier to understand.

## Key findings

- Proof confirms the BMV conjecture.
- Simplifies understanding of the original proof.
- Highlights key insights and intuition behind the proof.

## Abstract

The Bessis-Moussa-Villani conjecture states that the trace of $\exp(A-tB)$ is, as a function of the real variable $t$, the Laplace transform of a positive measure, where $A$ and $B$ are respectively a hermitian and positive semi-definite matrix. The long standing conjecture was recently proved by Stahl and streamlined by Eremenko. We report on a more concise yet self-contained version of the proof.