Sums of Hermitian Squares as an Approach to the BMV Conjecture
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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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On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction
The refined BMV upper bound fails for 3x3 PSD matrices, explained via non-canonical common parts, and a pinching refinement is proposed and proved for the n,2 case as a sandwich inequality.
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