Sums of Hermitian Squares as an Approach to the BMV Conjecture

2008/02/08 by Sabine Burgdorf, Burgdorf, Sabine · 1 citation
Mathematics · Computer Science · #Advanced Algebra and Geometry #Polynomial and algebraic computation #Matrix Theory and Algorithms

paper · doi:10.48550/arxiv.0802.1153

Abstract

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa-Villani (BMV) conjecture that all coefficients of the polynomial p(t)=Tr[(A+tB)^m], where A and B are positive semidefinite matrices of the same size and m an arbitrary integer, are nonnegative. The coefficient of t^k is the trace of S_m,k(A,B), which is the sum of all words of length m in the letters A and B in which B appears exactly k times. We consider the case k=4 and show that S_m,4(A,B) is a sum of hermitian squares and commutators. In particular, the trace of S_m,4(A,B) is nonnegative.

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