Complete monotonicity for inverse powers of some combinatorially defined polynomials

Alexander D. Scott Mathematical Institute, University of Oxford Alan D. Sokal Department of Physics, New York University

Functional Analysis mathscidoc:1701.12007

Acta Mathematica, 213, (2), 323-392, 2013.1
We prove the complete monotonicity on $${(0, \infty)^{n}}$$ 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 $${P^{-\beta}}$$ 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 $${P^{-\beta}}$$ for some $${\beta > 0}$$ 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.
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
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  title={Complete monotonicity for inverse powers of some combinatorially defined polynomials},
  author={Alexander D. Scott, and Alan D. Sokal},
  booktitle={Acta Mathematica},
Alexander D. Scott, and Alan D. Sokal. Complete monotonicity for inverse powers of some combinatorially defined polynomials. 2013. Vol. 213. In Acta Mathematica. pp.323-392.
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