Sum-of-Squares Results for Polynomials Related to the BessisMoussaVillani Conjecture
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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 RX,Y, where X2=A and Y2=B, for all even values of m and k with 6km-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 RA,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. Springer Science+Business Media, LLC 2010.