Complete monotonicity for inverse powers of some combinatorially defined polynomials

• 成果类型: Article
• 署名作者: Scott, Alexander D.; Sokal, Alan D.
• 署名单位: University of Oxford; New York University; University of London; University College London
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-014-0121-6
• 发表日期: 2014
• 页码: 323-392
• 关键词: half-plane property; infinitely divisible matrices; multivariate sequences; integral-representation; semi-groups; Positivity; asymptotics; distributions; variables; THEOREMS

摘要

We prove the complete monotonicity on 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 SzegA 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 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 for some 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.