Sums of Hermitian Squares as an Approach to the BMV Conjecture
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.
arXiv comment9 pages, grammatical corrections, typos added, new references
References in corpus (9)
- Connes' embedding conjecture and sums of hermitian squares
- Equivalent forms of the Bessis-Moussa-Villani conjecture
- Sums of hermitian squares and the BMV conjecture
- Proof of the cases of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture
- Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture
- On D. Haegele's approach to the Bessis-Moussa-Villani conjecture
- Advances on the Bessis-Moussa-Villani Trace Conjecture
- Asymptotic Positivity of Hurwitz Product Traces: Two Proofs
- On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices