Asymptotic Positivity of Hurwitz Product Traces

TL;DR

This paper proves that the coefficients of the polynomial trace of (A + tB)^m are asymptotically positive for large m, supporting the Bessis-Moussa-Villani conjecture in a specific case.

Contribution

It establishes asymptotic positivity of polynomial coefficients in the trace expansion for positive Hermitian matrices, advancing understanding of the Bessis-Moussa-Villani conjecture.

Findings

Coefficients become positive for all sufficiently large m

Supports the Bessis-Moussa-Villani conjecture asymptotically

Provides bounds depending on matrices A, B, and coefficient index

Abstract

Consider the polynomial $tr (A + tB)^m$ in $t$ for positive hermitian matrices $A$ and $B$ with $m \in \N$. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of $AB \neq 0$. More precisely, we show that the $k$-th coefficient is positive for all integer $m \geq m_0$, where $m_0$ depends on $A$, $B$ and $k$.