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Sum-of-squares results for polynomials related to the Bessis-Moussa-Villani conjecture. (English) Zbl 1197.81145

Let \(S_{m,k}(A,B)\) denote the sum of all words of length \(m\) in A and B having \(k\) letters equal to B and \(m-k\) equal to A. Then the coefficient of \(t^k\) in the polynomial \(p(t)= \text{Tr}((A+tB)^m)\) is equal to the trace of \(S_{m,k}(A,B)\), and the Lieb-Seirenger reformulation of the Bessis-Moussa-Villani conjecture is that this trace is always nonnegative. The authors show that the polynomial \(S_{m,k}(A,B)\), that is, the sum of all words in noncommuting variables A and B, is not equal to a sum of commutators and Hermitian squares in the algebra \(R\langle X,Y\rangle \), where \(X^2=A\) and \(Y^2=B\), for all even values of \(m\) and \(k\) with \(6\leq k\leq m-10,\) and also for \((m,k)=(12,6)\). This leaves only the case \((m,k)=(16,8)\) open. These results eliminate the possibility of using “descent+sum-of-squares” to prove the Bessis-Moussa-Villani conjecture.

MSC:

81R15 Operator algebra methods applied to problems in quantum theory
46T12 Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
82B05 Classical equilibrium statistical mechanics (general)
82B10 Quantum equilibrium statistical mechanics (general)

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References:

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