Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

TL;DR

This paper proves that the coefficients of the polynomial tr (A + tB)^m are asymptotically positive for positive Hermitian matrices, providing two different proofs for the nontrivial case where AB is nonzero.

Contribution

It establishes the asymptotic positivity of polynomial coefficients related to the Bessis-Moussa-Villani conjecture using complex analysis and combinatorial methods.

Findings

Coefficients become positive for all sufficiently large m.

Two distinct proofs: complex-analytic and combinatorial.

Results hold when AB ≠ 0.

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 - once complex-analytically, once combinatorially - that the $k$-th coefficient is positive for all integer $m \geq m_0$, where $m_0$ depends on $A$, $B$ and $k$.