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Principal minor sums of \((A + tB)^m\). (English) Zbl 1086.15506
Summary: The question is raised whether the sum of the \(k \times k\) principal minors of the titled matrix is a polynomial (in \(t\)) with positive coefficients, when \(A\) and \(B\) are positive definite. This would generalize a conjecture made by D. Bessis, P. Moussa, and M. Villani [J. Math. Phys. 16, 2318–2325 (1975; Zbl 0976.82501)], as stated by E. H. Lieb and R. Seiringer [Equivalent forms of the Bessis-Moussa-Villani conjecture, J. Stat. Phys. 115, 185–190 (2004)]. We give a variety of evidence for this further question, some of which is new.

MSC:
15A15 Determinants, permanents, traces, other special matrix functions
15B57 Hermitian, skew-Hermitian, and related matrices
15A90 Applications of matrix theory to physics (MSC2000)
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[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
[2] Johnson, C.R.; Hillar, C., Eigenvalues of words in two positive definite letters, SIAM J. matrix anal. appl., 23, 916-928, (2002) · Zbl 1007.68139
[3] Hillar, C.; 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) · Zbl 1126.15303
[4] Horn, R.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press New York · Zbl 0576.15001
[5] Lieb, E.H.; Seiringer, R., Equivalent forms of the bessis-moussa-villani conjecture, J. stat. phys., 115, 185-190, (2004) · Zbl 1157.81313
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