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Sums of Hermitian squares and the BMV conjecture. (English) Zbl 1158.15018

Summary: We show that all the coefficients of the polynomial \(\text{tr}\left((A+tB)^m\right)\in\mathbb{R}[t]\) are nonnegative whenever \(m\leq 13\) is a nonnegative integer and \(A\) and \(B\) are positive semidefinite matrices of the same size. This has previously been known only for \(m\leq 7\). The validity of the statement for arbitrary \(m\) has recently been shown to be equivalent to the Bessis–Moussa–Villani (BMV)-conjecture [D. Bessis, P. Moussa and M. Villani, J. Math. Phys. 16, 2318–2325 (1975; Zbl 0976.82501)] from theoretical physics. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.

MSC:

15A45 Miscellaneous inequalities involving matrices
90C22 Semidefinite programming
82B05 Classical equilibrium statistical mechanics (general)
15B48 Positive matrices and their generalizations; cones of matrices
16S36 Ordinary and skew polynomial rings and semigroup rings
46L35 Classifications of \(C^*\)-algebras

Citations:

Zbl 0976.82501
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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) · Zbl 0976.82501 · doi:10.1063/1.522463
[2] Burgdorf, S.: Sums of Hermitian squares as an approach to the BMV conjecture. Preprint, arXiv:0802.1153 · Zbl 1229.13020
[3] Choi, M.D., Lam, T.Y., Reznick, B.: Sums of squares of real polynomials. Proc. Symp. Pure Math. 58(2), 103–126 (1995) · Zbl 0821.11028
[4] Fleischhack, C.: Asymptotic positivity of Hurwitz product traces. Preprint, arXiv:0804.3665 · Zbl 1189.15008
[5] Friedland, S.: Remarks on BMV conjecture. Preprint, arXiv:0804.3948
[6] Hägele, D.: Proof of the cases p of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 127(6), 1167–1171 (2007) · Zbl 1117.82008 · doi:10.1007/s10955-007-9327-8
[7] Helton, J.W.: ”Positive” noncommutative polynomials are sums of squares. Ann. Math. (2) 156(2), 675–694 (2002) · Zbl 1033.12001 · doi:10.2307/3597203
[8] Helton, J.W., Putinar, M.: Positive polynomials in scalar and matrix variables, the spectraltheorem, and optimization. In: Operator Theory, Structured Matrices, and Dilations. Theta Ser. Adv. Math. vol. 7, pp. 229–306 (2007). arXiv:math/0612103 · Zbl 1199.47001
[9] Helton, J.W., Miller, R.L., Stankus, M.: NCAlgebra: A Mathematica package for doing non commuting algebra. Available from http://www.math.ucsd.edu/\(\sim\)ncalg/
[10] Hillar, C.J.: Advances on the Bessis-Moussa-Villani trace conjecture. Linear Algebra Appl. 426(1), 130–142 (2007) · Zbl 1126.15024 · doi:10.1016/j.laa.2007.04.005
[11] Hillar, C.J.: Sums of polynomial squares over totally real fields are rational sums of squares. Proc. Am. Math. Soc. (2008, to appear). arXiv:0704.2824 · Zbl 1163.12005
[12] 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(3–4), 781–789 (2005) · Zbl 1126.15303 · doi:10.1007/s10955-004-8829-x
[13] Klep, I., Schweighofer, M.: A Nichtnegativstellensatz for polynomials in noncommuting variables. Israel J. Math. 161(1), 17–27 (2007) · Zbl 1171.47060 · doi:10.1007/s11856-007-0070-2
[14] Klep, I., Schweighofer, M.: Sums of hermitian squares. Connes’ embedding problem and the BMV conjecture. Oberwolfach Rep. 4(1), 779–782 (2007)
[15] Klep, I., Schweighofer, M.: Connes’ embedding conjecture and sums of hermitian squares. Adv. Math. 217(4), 1816–1837 (2008) · Zbl 1184.46055 · doi:10.1016/j.aim.2007.09.016
[16] Landweber, P.S., Speer, E.R.: On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture. Preprint, arXiv:0711.0672 · Zbl 1175.82006
[17] Le Couteur, K.J.: Representation of the function Tr(exp(A B)) as a Laplace transform with positive weight and some matrix inequalities. J. Phys. A 13(10), 3147–3159 (1980) · Zbl 0446.44002 · doi:10.1088/0305-4470/13/10/012
[18] Lieb, E.H., Seiringer, R.: Equivalent forms of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 115(1–2), 185–190 (2004) · Zbl 1157.81313 · doi:10.1023/B:JOSS.0000019811.15510.27
[19] Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, pp. 284–289 (2004). http://control.ee.ethz.ch/\(\sim\)joloef/yalmip.php
[20] Moussa, P.: On the representation of Tr(e (A B)) as a Laplace transform. Rev. Math. Phys. 12(4), 621–655 (2000) · Zbl 0976.82027
[21] Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. Ser. Discrete Math. Theor. Comput. Sci. 60, 83–99 (2003) · Zbl 1099.13516
[22] Peyrl, H., Parrilo, P.A.: A Macaulay 2 package for computing sum of squares decompositions of polynomials with rational coefficients. In: Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, London, Ontario, Canada, pp. 207–208 (2007)
[23] Sturm, J.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11/12(1–4), 625–653 (1999) · Zbl 0973.90526 · doi:10.1080/10556789908805766
[24] Todd, M.J.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001) · Zbl 1105.65334 · doi:10.1017/S0962492901000071
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