Abstract
We show that the polynomial S m,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 X 2=A and Y 2=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 (S m,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 S m,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 (S m,4(A,B))≥0 holds for all Hermitian matrices A and B, for these values of m.
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B. Collins research supported in part by NSERC grant RGPIN/341303-2007 and by the Granma and Galoisint ANR grants.
K.J. Dykema research supported in part by NSF grants DMS-0600814 and DMS-0901220.
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Collins, B., Dykema, K.J. & Torres-Ayala, F. Sum-of-Squares Results for Polynomials Related to the Bessis–Moussa–Villani Conjecture. J Stat Phys 139, 779–799 (2010). https://doi.org/10.1007/s10955-010-9959-y
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DOI: https://doi.org/10.1007/s10955-010-9959-y