Polya-Schur master theorems for circular domains and their boundaries

Details Polya-Schur master theorems for circular domains and their boundaries Polya-Schur master theorems for circular domains and their boundaries

2006-07-18

arxiv.org/abs/math/0607416v6

Ann. of Math. (2) 170 (2009), no. 1, 465--492

Mathematics

Complex Variables

Final version, to appear in Ann. of Math.; 23 pages, no figures, LaTeX2e

Scientific paper

10.4007/annals.2009.170.465

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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