Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

Details Asymptotic Positivity of Hurwitz Product Traces: Two Proofs Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

2008-10-31

arxiv.org/abs/0811.0030v1

Physics

Mathematical Physics

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

Scientific paper

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 $k$.

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