Paper detail

Polya-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 $Ω\subseteq \mathbb{C}$ for arbitrary closed circular domains $Ω$ (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 $Ω=\mathbb{R}$ our results settle open questions that go back to Laguerre and Pólya-Schur.