Łojasiewicz-type inequalities in complex analysis
type: Item
instance of: scholarly article
title: Łojasiewicz-type inequalities in complex analysis
zbMATH Open document ID: 1409.32004
author: Rafał Pierzchała
published in: Journal d'Analyse Mathématique
publication date: 2018-10-02T00:00:00Z
review text: Let \(K \subseteq \mathbb C^N\) be a non-empty compact set. The Siciak extremal function \(\Phi_K\) is defined by \[ \Phi_K(z) := \sup \Big\{ |p(z)|^{1/\deg p} : p \in \mathbb C[z],\, \deg p >0,\, \sup_{z \in K} |p(z)| \leq 1\Big\}, \quad z \in \mathbb C^N. \] It is an important tool in approximation theory, complex analysis and pluripotential theory. The compact set \(K \subseteq \mathbb C^N\) is said to satisfy the Łojasiewicz-Siciak (ŁS) condition if there exist \(\eta, \kappa >0\) and an open neighborhood \(U\) of \(K\) in \(\mathbb C^N\) such that \[ \Phi_K(z) \geq 1 + \eta \, \text{dist} (z,K)^\kappa \quad \text{ for all } z \in U. \] In particular, if \(K\) satisfies the (ŁS) condition then it is polynomially convex. In the paper under review the author provides conditions under which the (ŁS) condition is preserved by taking preimages or images of holomorphic mappings. As a consequence many new examples of sets having the (ŁS) property are given. The paper continues previous work on the (ŁS) condition of same the author started in [J. Math. Anal. Appl. 430, No. 2, 755--776 (2015; Zbl 1331.32017)] using techniques from subanalytic geometry, most notably the Łojasiewicz inequality.
reviewed by: Armin Rainer
Mathematics Subject Classification ID: 32B20
Mathematics Subject Classification ID: 32U05
zbMATH DE Number: 6946525
zbMATH Keywords: Siciak extremal function
zbMATH Keywords: Łojasiewicz-Siciak condition
zbMATH Keywords: subanalytic sets
MaRDI profile type: Publication
full work available at URL: s11854-018-0016-0
OpenAlex ID: W2789619108
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DOI: 10.1007/S11854-018-0016-0
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description: scientific article; zbMATH DE number 6946525