Abstract
It is shown that the polynomial
p(t) = Tr[(A+tB)m]
has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa, and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to p(t) having positive coefficients for any m and any two n-by-n positive definite matrices. We show that, generally, the question in the real case reduces to that of singular A and B, and this is a key part of our proof.
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Hillar, C.J., Johnson, C.R. On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices. J Stat Phys 118, 781–789 (2005). https://doi.org/10.1007/s10955-004-8829-x
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DOI: https://doi.org/10.1007/s10955-004-8829-x