Pólya-Schur master theorems for circular domains and their boundaries


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.


Julius Borcea

Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden

Petter Brändén

Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden