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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)
##### Keywords:
positive definite matrix; principal minors
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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 [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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