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

- 55 Downloads
- 2 Citations

## 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**〈*X*,*Y*〉, 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 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 **R**〈*A*,*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*.

## Keywords

BMV conjecture Hermitian squares## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bessis, D., Moussa, P., Villani, M.: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys.
**16**, 2318–2325 (1975) CrossRefMathSciNetADSGoogle Scholar - 2.Burgdorf, S.: Sums of Hermitian squares as an approach to the BMV conjecture. Linear Multilinear Algebra (to appear) Google Scholar
- 3.Hägele, D.: Proof of the cases
*p*≤7 of the Lieb–Seiringer formulation of the Besis–Moussa–Villani conjecture. J. Stat. Phys.**127**, 1167–1171 (2007) MATHCrossRefMathSciNetADSGoogle Scholar - 4.Hillar, C.J.: Advances on the Bessis–Moussa–Villani trace conjecture. Linear Algebra Appl.
**426**, 130–142 (2007) MATHCrossRefMathSciNetGoogle Scholar - 5.Hillar, C.J., Johnson, C.R.: On the positivity of the coefficients of a certain polynomial defined by two positive definite matrices. J. Stat. Phys.
**118**, 781–789 (2005) MATHCrossRefMathSciNetADSGoogle Scholar - 6.Klep, I., Schweighofer, M.: Sums of Hermitian squares and the BMV conjecture. J. Stat. Phys.
**133**, 739–760 (2008) MATHCrossRefMathSciNetADSGoogle Scholar - 7.Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of Hermitian squares. Adv. Math.
**217**, 1816–1837 (2008) CrossRefMathSciNetGoogle Scholar - 8.Landweber, P.S., Speer, E.R.: On D. Hägele’s approach to the Bessis–Moussa–Villani conjecture. Linear Algebra Appl.
**431**, 1317–1324 (2009) MATHCrossRefMathSciNetGoogle Scholar - 9.Lieb, E.H., Seiringer, R.: Equivalent forms of the Bessis–Moussa–Villani conjecture. J. Stat. Phys.
**115**, 185–190 (2004) MATHCrossRefMathSciNetADSGoogle Scholar - 10.Wolfram Research, Inc.: Mathematica Version 7.0. Wolfram Research, Inc., Champaign (2008) Google Scholar