Journal of Statistical Physics

, Volume 139, Issue 5, pp 779–799 | Cite as

Sum-of-Squares Results for Polynomials Related to the Bessis–Moussa–Villani Conjecture

  • Benoît Collins
  • Kenneth J. Dykema
  • Francisco Torres-Ayala


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 RX,Y〉, where X 2=A and Y 2=B, for all even values of m and k with 6≤km−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 RA,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.


BMV conjecture Hermitian squares 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Benoît Collins
    • 1
    • 2
  • Kenneth J. Dykema
    • 3
  • Francisco Torres-Ayala
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.CNRS, Department of MathematicsLyon 1 Claude Bernard UniversityVileurbanneFrance
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

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