We show that the polynomial Sm,k(A,B), that is the sum of all words in noncommuting variables A and B having length m and exactly k letters equal to B, is not equal to a sum of commutators and Hermitian squares in the algebra R〈X,Y〉, where X2=A and Y2=B, for all even values of m and k with 6≤k≤m−10, and also for (m,k)=(12,6). This leaves only the case (m,k)=(16,8) open. This topic is of interest in connection with the Lieb–Seiringer formulation of the Bessis–Moussa–Villani conjecture, which asks whether Tr (Sm,k(A,B))≥0 holds for all positive semidefinite matrices A and B. These results eliminate the possibility of using “descent + sum-of-squares” to prove the BMV conjecture.
We also show that Sm,4(A,B) is equal to a sum of commutators and Hermitian squares in R〈A,B〉 when m is even and not a multiple of 4, which implies Tr (Sm,4(A,B))≥0 holds for all Hermitian matrices A and B, for these values of m.
BMV conjecture Hermitian squares
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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)