On D. Haegele's approach to the Bessis-Moussa-Villani conjecture

TL;DR

This paper explores an extension of Haegele's method to address the Bessis-Moussa-Villani conjecture, analyzing its effectiveness for various matrix parameters and providing complete results when either parameter is odd.

Contribution

It introduces a natural extension of Haegele's approach and determines its success for specific cases of p and r, advancing understanding of the conjecture.

Findings

Method is successful when either p or r is odd.

Complete determination of success cases for these parameter values.

Provides insights into the structure of the conjecture for positive semidefinite matrices.

Abstract

The reformulation of the Bessis-Moussa-Villani conjecture given by Lieb and Seiringer asserts that the coefficient of t^r in the polynomial Trace[(A+tB)^p], with A and B positive semidefinite matrices, is nonnegative for all p and r. We propose a natural extension of a method of attack on this problem due to Haegele, and investigate for what values of p and r the method is successful, obtaining a complete determination when either p or r is odd.