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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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