Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture

Details Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture

2009-05-04

arxiv.org/abs/0905.0420v3

J. Stat. Phys. 139 (2010), 779-799

Mathematics

Rings and Algebras

21 pages. In the second version, pictures have been added and we point out that one of our results was previously obtained by

Scientific paper

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 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 the trace of S_{m,k}(A,B)) is nonnegative 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 when m is even and not a multiple of 4, which implies that the trace of S_{m,4}(A,B) is nonnegative for all Hermitian matrices A and B, for these values of m.

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