Sums of Hermitian Squares as an Approach to the BMV Conjecture

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.

9 pages, grammatical corrections, typos added, new references

References in corpus (4)

• Proof of the cases $p\le 7$ of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani 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
• Asymptotic Positivity of Hurwitz Product Traces: Two Proofs

Cited by in corpus (2)

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