The Lee–Yang and Pólya–Schur programs. II: Theory of stable polynomials and applications. (English) Zbl 1177.47041
This paper is a continuation of J.Borcea and P.Brändén, [“The Lee-Yang and Pólya-Schur programs.I: Linear operators preserving stability”, Invent.Math.177, No.3, 541–569 (2009; Zbl 1175.47032)].
Let \(K=\mathbb R\) or \(=\mathbb C\). If \(n\) is a positive integer and \(\Omega\subset\mathbb K^n\), a polynomial \(f\in\mathbb K[z_1,\dots,z_n]\) is said to be \(\Omega\)-stable if \(f(z_1,\dots,z_n)\not=0\) whenever \((z_1,\dots,z_n)\in\Omega\). Denote the class of all \(K\)-polynomials in the variables \(z_1,\dots,z_n\) by \(K[z_1,\dots,z_n]\). Also, for \(\kappa\in\mathbb N^n\) let \(K_\kappa[z_1,\dots,z_n] = \{f\in K[z_1,\dots,z_n] : \deg_{z_i}(f)\leq \kappa_i\;\mathrm{for}\;\mathrm{each}\; 1\leq i\leq n\}\), where \(\deg_{z_i}(f)\) is the degree of \(f\) in \(z_i\).
The authors investigate the following problems, which have been previously investigated in a large variety of special cases:
Problem 1: Characterize all linear operators \[ T:\mathbb K_\kappa[z_1,\dots,z_n]\to \mathbb K[z_1,\dots,z_n] \] that preserve \(\Omega\)-stability, where \(\Omega\) is a prescribed subset of \(\mathbb K^n\) and \(\kappa\in\mathbb N^n\).
Problem 2: Characterize all linear operators \[ T:\mathbb K[z_1,\dots,z_n]\to \mathbb K[z_1,\dots,z_n] \] that preserve \(\Omega\)-stability, where \(\Omega\) is a prescribed subset of \(\mathbb K^n\).
In this second part, the authors extend their general approach to a variety of topics. The table of contents is as follows: (1.) Symmetrization Procedures. (2.) Grace–Walsh–Szegő Coincidence Theorem. (3) Master Composition Theorems. (4.) Hard Pólya–Schur Theory: Bounded Degree Multiplier Sequences. (5.) Multivariate Apolarity. (6.) Hard Lieb–Sokal Lemmas. (7.) Transcendental Symbols and the Weyl Algebra. (8.) Applications: Recovering Lee–Yang and Heilmann–Lieb Type Theorems.
Let \(K=\mathbb R\) or \(=\mathbb C\). If \(n\) is a positive integer and \(\Omega\subset\mathbb K^n\), a polynomial \(f\in\mathbb K[z_1,\dots,z_n]\) is said to be \(\Omega\)-stable if \(f(z_1,\dots,z_n)\not=0\) whenever \((z_1,\dots,z_n)\in\Omega\). Denote the class of all \(K\)-polynomials in the variables \(z_1,\dots,z_n\) by \(K[z_1,\dots,z_n]\). Also, for \(\kappa\in\mathbb N^n\) let \(K_\kappa[z_1,\dots,z_n] = \{f\in K[z_1,\dots,z_n] : \deg_{z_i}(f)\leq \kappa_i\;\mathrm{for}\;\mathrm{each}\; 1\leq i\leq n\}\), where \(\deg_{z_i}(f)\) is the degree of \(f\) in \(z_i\).
The authors investigate the following problems, which have been previously investigated in a large variety of special cases:
Problem 1: Characterize all linear operators \[ T:\mathbb K_\kappa[z_1,\dots,z_n]\to \mathbb K[z_1,\dots,z_n] \] that preserve \(\Omega\)-stability, where \(\Omega\) is a prescribed subset of \(\mathbb K^n\) and \(\kappa\in\mathbb N^n\).
Problem 2: Characterize all linear operators \[ T:\mathbb K[z_1,\dots,z_n]\to \mathbb K[z_1,\dots,z_n] \] that preserve \(\Omega\)-stability, where \(\Omega\) is a prescribed subset of \(\mathbb K^n\).
In this second part, the authors extend their general approach to a variety of topics. The table of contents is as follows: (1.) Symmetrization Procedures. (2.) Grace–Walsh–Szegő Coincidence Theorem. (3) Master Composition Theorems. (4.) Hard Pólya–Schur Theory: Bounded Degree Multiplier Sequences. (5.) Multivariate Apolarity. (6.) Hard Lieb–Sokal Lemmas. (7.) Transcendental Symbols and the Weyl Algebra. (8.) Applications: Recovering Lee–Yang and Heilmann–Lieb Type Theorems.
Reviewer: Vania Mascioni (Muncie)
MSC:
| 47B38 | Linear operators on function spaces (general) |
| 05A15 | Exact enumeration problems, generating functions |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 32A60 | Zero sets of holomorphic functions of several complex variables |
| 46E22 | Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
Keywords:
stable polynomial; linear operator; multivariate polynomial; open circular domain; Polya-Schur; Lee-Yang; multivariate apolarityCitations:
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