Advances on the Bessis–Moussa–Villani trace conjecture

A long-standing conjecture asserts that the polynomialp(t)=Tr[(A+tB)m]p(t)=Tr[(A+tB)m]has nonnegative coefficients whenever m is a positive integer and A and B are any two n × n positive semidefinite Hermitian matrices. The conjecture arises from a question raised by Bessis et al. [D. Bessis, P. Moussa, M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys. 16 (1975) 2318–2325] in connection with a problem in theoretical physics. Their conjecture, as shown recently by Lieb and Seiringer, is equivalent to the trace positivity statement above. In this paper, we derive a fundamental set of equations satisfied by A and B that minimize or maximize a coefficient of p(t). Applied to the Bessis–Moussa–Villani (BMV) conjecture, these equations provide several reductions. In particular, we prove that it is enough to show that (1) it is true for infinitely many m, (2) a nonzero (matrix) coefficient of (A + tB)m always has at least one positive eigenvalue, or (3) the result holds for singular positive semidefinite matrices. Moreover, we prove that if the conjecture is false for some m, then it is false for all larger m. Finally, we outline a general program to settle the BMV conjecture that has had some recent success.