2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/230720Lieb 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 referencesFunctional AnalysisMathematical PhysicsOperator Algebras11E25; 13J30; 15A90; 15A45text