2008/09/18 by Julius Borcea, Petter Brändén · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Function (biology) #Generalization #Holomorphic and Operator Theory #Multivariate statistics #Polynomial and algebraic computation #Property (philosophy) #Series (stratigraphy) #Univariate #cond-mat.stat-mech #math-ph #math.CO #math.CV #math.MP #msc:05A15 #msc:05C70 #msc:30C15 #msc:32A60 #msc:46E22 #msc:47B38 #msc:82B20 #msc:82B26
paper · pdf · doi:10.1002/cpa.20295
published as Comm. Pure Appl. Math. 62 (2009), no. 12, 1595-1631 · 32 pages
arxiv created 2008/09/18 · openalex publication_date 2009/08/11 · arxiv updated 2012/04/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee‐Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity. © 2009 Wiley Periodicals, Inc.