Acta Mathematica

Complete monotonicity for inverse powers of some combinatorially defined polynomials

Alexander D. Scott and Alan D. Sokal

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We prove the complete monotonicity on (0,)n for suitable inverse powers of the spanning-tree polynomials of graphs and, more generally, of the basis generating polynomials of certain classes of matroids. This generalizes a result of Szegő and answers, among other things, a long-standing question of Lewy and Askey concerning the positivity of Taylor coefficients for certain rational functions. Our proofs are based on two ab-initio methods for proving that P-β is completely monotone on a convex cone C: the determinantal method and the quadratic-form method. These methods are closely connected with harmonic analysis on Euclidean Jordan algebras (or equivalently on symmetric cones). We furthermore have a variety of constructions that, given such polynomials, can create other ones with the same property: among these are algebraic analogues of the matroid operations of deletion, contraction, direct sum, parallel connection, series connection and 2-sum. The complete monotonicity of P-β for some β>0 can be viewed as a strong quantitative version of the half-plane property (Hurwitz stability) for P, and is also related to the Rayleigh property for matroids.

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Acta Math., Volume 213, Number 2 (2014), 323-392.

Received: 11 January 2013
Revised: 13 November 2013
First available in Project Euclid: 30 January 2017

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Zentralblatt MATH identifier

Primary: 05C31 (Primary)
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A20: Combinatorial inequalities 05B35: Matroids, geometric lattices [See also 52B40, 90C27] 05C05: Trees 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 05E99: None of the above, but in this section 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14] 15B33: Matrices over special rings (quaternions, finite fields, etc.) 15B57: Hermitian, skew-Hermitian, and related matrices 17C99: None of the above, but in this section 26A48: Monotonic functions, generalizations 26B25: Convexity, generalizations 26C05: Polynomials: analytic properties, etc. [See also 12Dxx, 12Exx] 32A99: None of the above, but in this section 43A85: Analysis on homogeneous spaces 44A10: Laplace transform 60C05: Combinatorial probability 82B20 (Secondary)

complete monotonicity positivity inverse power fractional power polynomial spanning-tree polynomial basis generating polynomial elementary symmetric polynomial matrixtree theorem determinant quadratic form half-plane property Hurwitz stability Rayleigh property Bernstein–Hausdorff–Widder theorem Laplace transform harmonic analysis symmetric cone Euclidean Jordan algebra Gindikin–Wallach set

2014 © Institut Mittag-Leffler


Scott, Alexander D.; Sokal, Alan D. Complete monotonicity for inverse powers of some combinatorially defined polynomials. Acta Math. 213 (2014), no. 2, 323--392. doi:10.1007/s11511-014-0121-6.

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