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 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.

LaTeX2e, 70 pages (v2) or 82 pages (v3). Version 2 (accepted for publication in Acta Mathematica) is significantly reorganized at the suggestion of a referee; also, Appendix A is deleted to save space. Version 3 is the expanded version for the arXiv: it contains Appendices A and B that are not included, due to space constraints, tn the version (v2) that will be published in Acta Mathematica

References in corpus (1)

• Grassmann Integral Representation for Spanning Hyperforests

Cited by in corpus (3)

• Exponential Varieties
• A combinatorial proof of the Gaussian product inequality beyond the MTP${}_2$ case
• On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices