On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices

TL;DR

This paper proves that the polynomial p(t) = Tr[(A+tB)^6] has positive coefficients when A and B are 3x3 positive definite matrices, addressing a specific case related to a broader conjecture in physics.

Contribution

It establishes the positivity of polynomial coefficients for a particular case (m=6, 3x3 matrices), which was previously unproven and extends understanding of the conjecture.

Findings

Proves positivity for m=6, 3x3 matrices

Reduces the general problem to the case of singular matrices

Addresses a conjecture linked to physics and matrix analysis

Abstract

It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to $p(t)$ having positive coefficients for any $m$ and any two $n$-by-$n$ positive definite matrices. We show that, generally, the question in the real case reduces to that of singular $A$ and $B$, and this is a key part of our proof.