# Complete monotonicity for inverse powers of some combinatorially defined polynomials

Scott, A
Sokal, A

1 January 2014

Acta Mathematica

## Last Updated:

2020-07-19T19:17:18.62+01:00

2

213

## DOI:

10.1007/s11511-014-0121-6

323-392

## abstract:

We prove the complete monotonicity on (0,∞)n(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−βP-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−βP-β for some β&gt;0β&gt;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.

492535

Submitted

Journal Article