Local polynomial convexity of certain graphs in \(C^{3}\)
type: Item
instance of: scholarly article
title: Local polynomial convexity of certain graphs in \(C^{3}\)
zbMATH Open document ID: 1192.32009
author: Phuong Chi Kieu
author: Nguyen Quang Dieu
published in: Michigan Mathematical Journal
publication date: 2009-12-15T00:00:00Z
review text: The aim of the authors is to prove the following theorem: Let \(f\) be a \(C^{2}\) smooth function on a neighborhood of \(0\in\mathbb{C}\). Assume that there exist a holomorphic function \(g\) defined near \(0\in\mathbb{C}\) and a constant \(\lambda \in (0,1)\) satisfying the following conditions: (i) \(|f|^{2} \leq Re(fg)\); (ii) \(g(0)=0\); (iii) \(|g\triangle \overline{f}|<\lambda |\frac{\partial f}{\partial \overline{z}}|^{2}+Re(f\triangle \overline{f}\) for every \(z\neq 0\). Then \(\Gamma_{f}\) is locally polynomially convex at the origin in \(0\in\mathbb{C}^{2}\). Furthermore, there exists an \(r>0\) small enough such that a continuous function on \(X_{r}:=\Gamma_{f}\cap \overline{B}(0,r)\) can be approximated uniformly by polynomials. Few corollaries of this theorem show that the local polynomial convexity of totally real graphs is well-behaved under non-degenerate holomorphic mappings.
reviewed by: Gabriela Cristescu
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32V20
Mathematics Subject Classification ID: 46J10
zbMATH DE Number: 5648488
zbMATH Keywords: polynomial convexity
zbMATH Keywords: local polynomial convexity
zbMATH Keywords: totally real
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1250169073
OpenAlex ID: W2012109477
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DOI: 10.1307/MMJ/1250169073
description: scientific article; zbMATH DE number 5648488