Pólya-Schur master theorems for circular domains and their boundaries
Annals of mathematics, Tome 170 (2009) no. 1, pp. 465-492
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 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 $\Omega=\mathbb{R}$ our results settle open questions that go back to Laguerre and Pólya-Schur.
@article{10_4007_annals_2009_170_465,
author = {Julius Borcea and Petter Br\"and\'en},
title = {P\'olya-Schur master theorems for circular domains and their boundaries},
journal = {Annals of mathematics},
pages = {465--492},
year = {2009},
volume = {170},
number = {1},
doi = {10.4007/annals.2009.170.465},
mrnumber = {2521123},
zbl = {1184.30004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2009.170.465/}
}
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Julius Borcea; Petter Brändén. Pólya-Schur master theorems for circular domains and their boundaries. Annals of mathematics, Tome 170 (2009) no. 1, pp. 465-492. doi: 10.4007/annals.2009.170.465
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