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  ރ for arbitrary closed circular domains (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 D ޒ our results settle open questions that go back to Laguerre and Pólya-Schur.
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