On D. Hägele’s approach to the Bessis–Moussa–Villani conjecture

The reformulation of the Bessis–Moussa–Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient αm,k(A,B) of tk in the polynomial Tr(A+tB)m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.