The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

Abstract

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Pólya-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.

32 pages

References in corpus (4)

• Polynomials with the half-plane property and matroid theory
• Polya-Schur master theorems for circular domains and their boundaries
• Partition function zeros at first-order phase transitions: A general analysis
• Partition function zeros at first-order phase transitions: Pirogov-Sinai theory

Cited by in corpus (12)

• Obstructions to determinantal representability
• The Ising Partition Function: Zeros and Deterministic Approximation
• Location of zeros for the partition function of the Ising model on bounded degree graphs
• Matrix positivity preservers in fixed dimension. I
• Phase transitions with four-spin interactions
• On values of the Bessel function for generic representations of finite general linear groups
• Lee-Yang zeros in the Rydberg atoms
• The Newton polytope and Lorentzian property of chromatic symmetric functions
• Spaces of Lorentzian and real stable polynomials are Euclidean balls
• Linear finite difference operators preserving Laguerre-Pólya class
• Hermite-Poulain theorems for linear finite difference operators
• Lorentzian polynomials and the independence sequences of graphs