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 M\"obius 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 P\'olya-Schur.
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