Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

Abstract

Consider the polynomial $tr(A+tB{)}^m$ in $t$ for positive hermitian matrices $A$ and $B$ with $m\in \mathbb{N}$. 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 $AB\ne 0$. More precisely, we show - once complex-analytically, once combinatorially - that the $k$-th coefficient is positive for all integer $m\ge m_0$, where $m_0$ depends on $A$, $B$ and $k$.

21 pages, LaTeX; merges articles 0804.3665 [CF] and 0804.3948 [SF]

References in corpus (4)

• Proof of the cases $p\le 7$ of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture
• Asymptotic Positivity of Hurwitz Product Traces
• Remarks on BMV conjecture
• Sums of Hermitian Squares as an Approach to the BMV Conjecture

Cited by in corpus (1)

• Sums of Hermitian Squares as an Approach to the BMV Conjecture