Sums of hermitian squares as an approach to the BMV conjecture

摘要

Lieb and Seiringer stated in their reformulation of the Bessis-Moussa- Villani conjecture that all coefficients of the polynomial p(t)=tr((A+trB)~m) are non-negative whenever A and B are any two positive semidefinite matrices of the same size. We will show that for all m ∈ N the coefficient of t4 in p(t) is non-negative, using a connection to sums of Hermitian squares of non-commutative polynomials which has been established by Klep and Schweighofer. This implies by a well-known result of Hillar that the coefficients of t~k are non-negative for 0≤k≤4, and by symmetry as well for m≥k≥m-4.