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arXiv · 0905.0420

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

Benoit Collins·Kenneth J. Dykema·Francisco Torres-Ayala
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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 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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Benoit Collins, Kenneth J. Dykema, Francisco Torres-Ayala. 2009-05-04. Sum--of--squares results for polynomials related to the Bessis--Moussa--Villani conjecture. https://doi.org/10.1007/s10955-010-9959-y

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