Local polynomial convexity of graphs of functions in several variables
type: Item
instance of: scholarly article
title: Local polynomial convexity of graphs of functions in several variables
zbMATH Open document ID: 1327.32021
author: Kieu Phuong Chi
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-09-25T00:00:00Z
review text: The author studies the local polynomial convexity of graphs of smooth functions of several complex variables. The main result is the following theorem. Let \(U\) be an open neighborhood of \(0\in\mathbb C^n\) and let \(f\in\mathcal C^2(U,\mathbb C)\). Assume that there exists a \(g\in\mathcal C^2(U,\mathbb C)\) such that: {\parindent=8mm \begin{itemize} \item[(i)] \(|f|^2\leq\text{Re}(fg)\), \item [(ii)] there exists a \(p\in\mathbb N\) such that \(|g(z)|\leq|z|^p\) for \(z\in U\), and the function \((z,w)\longmapsto|w|^2-\text{Re}(wg(z))\) is plurisubharmonic in a neighborhood of \(0\in\mathbb C^{n+1}\), \item [(iii)] \(\Big|\frac{\partial f}{\partial\overline z}(z)\overline w\Big|^2+\text{Re}(f(z)L_{\overline f}(z)(w) )>|g(z)L_{\overline f}(z)(w)|\) for all \(z\in U\setminus\{0\}\) and \(w\in(\mathbb C\setminus\{0\})^n\). \end{itemize}} Then the graph \(\varGamma_f\) of \(f\) is polynomially convex at \(0\in\mathbb C^{n+1}\). Moreover, there exists an \(r>0\) such that each continuous function on \(\varGamma_f\cap B(0,r)\) can be approximated uniformly by polynomials.
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 6485380
zbMATH Keywords: polynomial convexity
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: polynomial approximation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2015.08.010
OpenAlex ID: W1059459655
cites work: Several Complex Variables and Banach Algebras
cites work: Surfaces with degenerate CR singularities that are locally polynomially convex
cites work: Polynomial approximation, local polynomial convexity, and degenerate CR singularities
cites work: Function algebras on disks. II
cites work: Local polynomial convexity of certain graphs in \(C^{3}\)
cites work: Function Algebras on Disks
cites work: Q3995750
cites work: Q3346564
cites work: Uniform approximation by polynomials in two functions
cites work: Eva Kallin's Lemma On Polynomial Convexity
cites work: Polynomial convexity
cites work: Approximation on a disk
cites work: Local polynomially convex hulls at degenerated CR singularities of surfaces in C^2
DOI: 10.1016/J.JMAA.2015.08.010
Recommended article: Local polynomial convexity of certain graphs in \(C^{3}\)
Recommended article: Q3773701
Recommended article: Local conicity and polyhedrality of convex sets
Recommended article: On nonconvexity of graphs of polynomials of several real variables
Recommended article: Local polynomial convexity of tangential unions of totally real graphs in \(\mathbb{C}^2\)
Recommended article: Polynomial convexity and Rossi's local maximum principle
Recommended article: Polynomials on locally K-convex spaces
Recommended article: On local convexity in graphs
Recommended article: Q4270235
Recommended article: Q3567739
description: scientific article; zbMATH DE number 6485380