Pólya-Schur master theorems for circular domains and their boundaries | Annals of Mathematics

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 Ω⊆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=ℝ our results settle open questions that go back to Laguerre and Pólya-Schur.

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数学年刊(Annals of Mathematics)

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