Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

2008/10/31 by Christian Fleischhack, Shmuel Friedland, Fleischhack, Christian +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Polynomial and algebraic computation #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.0811.0030

Abstract

Consider the polynomial tr (A + tB)^m in t for positive hermitian matrices A and B with m ∈ \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 ≠ 0. More precisely, we show - once complex-analytically, once combinatorially - that the k-th coefficient is positive for all integer m ≥ m₀, where m₀ depends on A, B and k.

Citations

• Remarks on BMV conjecture
• Asymptotic Positivity of Hurwitz Product Traces
• Sums of Hermitian Squares as an Approach to the BMV Conjecture

Related

• Asymptotic Positivity of Hurwitz Product Traces
• Advances on the Bessis-Moussa-Villani Trace Conjecture
• On D. Haegele's approach to the Bessis-Moussa-Villani conjecture
• On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices
• Sums of Hermitian Squares as an Approach to the BMV Conjecture