## Acta Mathematica

### Complete monotonicity for inverse powers of some combinatorially defined polynomials

#### Abstract

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.

#### Article information

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

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

https://projecteuclid.org/euclid.acta/1485801868

Digital Object Identifier
doi:10.1007/s11511-014-0121-6

Mathematical Reviews number (MathSciNet)
MR3286037

Zentralblatt MATH identifier
1304.05074

Rights

#### Citation

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. https://projecteuclid.org/euclid.acta/1485801868

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