Polya-Schur master theorems for circular domains and their boundaries

Abstract

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region $\Omega \subseteq \mathbb{C}$ for arbitrary closed circular domains $\Omega$ (i.e., images of the closed unit disk under a Möbius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for $\Omega =\mathbb{R}$ our results settle open questions that go back to Laguerre and Pólya-Schur.

Final version, to appear in Ann. of Math.; 23 pages, no figures, LaTeX2e

Cited by in corpus (16)

• The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications
• On the degree and half degree principle for symmetric polynomials
• Jensen Polynomials for the Riemann Xi Function
• Zeros of polynomials with four-term recurrence
• Hyperbolicity preservers and majorization
• Polynomials with rational generating functions and real zeros
• Hermite multiplier sequences and their associated operators
• A generalization of the Heine--Stieltjes theorem
• Complex zero strip decreasing operators
• The non-existence of cubic Legendre multiplier sequences
• Nuij type pencils of hyperbolic polynomials
• Linear finite difference operators preserving Laguerre-Pólya class
• Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets
• Hermite-Poulain theorems for linear finite difference operators
• The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra
• Parity of Polynomial Multiplier Sequences for the Chebyshev Basis