Complete monotonicity for inverse powers of some combinatorially defined polynomials

TL;DR

This paper proves complete monotonicity for inverse powers of certain combinatorial polynomials, extending previous results and connecting to harmonic analysis and matroid operations, with implications for stability and positivity properties.

Contribution

It introduces two methods for establishing complete monotonicity of inverse polynomial powers, generalizes known results, and develops constructions preserving this property for combinatorial polynomials.

Findings

Proves complete monotonicity for inverse powers of spanning-tree and basis generating polynomials.

Develops determinantal and quadratic-form methods linked to harmonic analysis.

Creates constructions that generate new polynomials with the same monotonicity property.

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 Szego 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.