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Pólya-Schur master theorems for circular domains and their boundaries. (English) Zbl 1184.30004
Let \(\Omega\subseteq \mathbb{C}\) be a closed circular domain (i.e., the image of the closed unit disk under a Möbius transformation) or its boundary. Denote by \(\pi(\Omega)\) the class of all (complex or real) univariate polynomials whose zeros lie in \(\Omega\), and by \(\pi_n(\Omega)\) the subclass of \(\pi(\Omega)\) consisting of polynomials of degree at most \( n\). In the paper, the following three problems are considered.
Problem 1. Characterize all linear transformations \( T:\pi(\Omega) \to \pi(\Omega)\cup \{0\}.\)
Problem 2. For \(n \in \mathbb{N}, \) describe all linear operators \( T:\pi_n(\Omega) \to \pi(\Omega)\cup \{0\}.\)
Problem 3. For \(n \in \mathbb{N}, \) describe all linear operators \( T:\pi_n(\Omega)\setminus \pi_{n-1}(\Omega) \to \pi(\Omega)\cup \{0\}.\)
The authors obtain exhaustive solutions of these problems. In particular, for \(\Omega = \{z \in \mathbb{C}: \operatorname{Im}(z)>0\}\), they give a description of all operators that preserve the property of polynomials to be stable. For \(\Omega =\mathbb{R}\), their results settle an open question on characterization of hyperbolicity preservers that goes back to Laguerre, Pólya, Schur. More precisely, they prove the following theorem.
Theorem. A linear operator \(T:\mathbb{R}[z] \to \mathbb{R}[z]\) is an operator of the kind \(T:\pi(\mathbb{R}) \to \pi(\mathbb{R})\cup \{0\}\) if and only if one of the following holds:
(a) \(T\) is of the form \(T(f)=\alpha (f)P+\beta (f)Q\), where \(\alpha, \beta: \mathbb{R}[z]\to \mathbb{R}\) are linear functionals, and \(P,Q \in \pi(\mathbb{R})\) have interlacing zeros;
(b) for all \(n \in \mathbb{N}\), either the polynomial \(T[(z+w)^n]\neq 0\) for \((z,w) \in \{(z,w) \in \mathbb{C}^2: \operatorname{Im} z >0,\operatorname{Im} w >0\}\) or \(T[(z+w)^n]\equiv 0\);
(c) for all \(n \in \mathbb{N}\), either the polynomial \(T[(z-w)^n]\neq 0\) for \((z,w) \in \{(z,w) \in \mathbb{C}^2: \operatorname{Im} z >0, \operatorname{Im} w >0\}\) or \(T[(z-w)^n]\equiv 0.\)

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
26C10 Real polynomials: location of zeros
30D15 Special classes of entire functions of one complex variable and growth estimates
47D03 Groups and semigroups of linear operators
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