Sums of Hermitian squares as an approach to the BMV conjecture
Sabine Burgdorf · Linear and Multilinear Algebra · 2010
Lieb and Seiringer stated in their reformulation of the Bessis–Moussa–Villani conjecture that all coefficients of the polynomial p(t) = tr((A +tr B) 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 t 4 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.