Complete monotonicity for inverse powers of some combinatorially defined polynomials

Journal article

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>0β>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.

Authors: Scott, A; Sokal, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Royal Swedish Academy of Sciences, Institut Mittag-Leffler
• Journal: Acta Mathematica
• Volume: 213
• Issue: 2
• Pages: 323-392
• Publication date: 2014-01-01
• Acceptance date: 2013-11-13
• DOI: 10.1007/s11511-014-0121-6
• EISSN: 1871-2509
• ISSN: 0001-5962
• Keywords: SBTMR