Polynomial convexity
type: Item
instance of: scholarly article
title: Polynomial convexity
zbMATH Open document ID: 1124.32007
published in: Progress in Mathematics
publication date: 2007-01-11T00:00:00Z
review text: Polynomial convexity is a topic playing an important role in questions related to in the theory of approximation of functions of several complex variables. The author presents some of the main achievements of the mathematical community in this domain. The first two chapters present some of the basic notions and tools on the subject: polynomial and rational convexity, theory of abstract uniform algebras and of plurisubharmonic functions, the Cauchy-Fantappiè integral which is a central ingredient in the proof of the Oka-Weil approximation theorem and some general properties of polynomially convex sets. As it turned out that understanding the structure of the polynomially convex hull \(\widehat{X}\) of a compact set \(X\) and the structure of \(\widehat{X} \backslash X\) (in a manifold or in the numeric space) is very important, one of the major directions of the book is the study of the one-dimensional examples where an analytic structure on these objects can be found. In the first chapter some examples of hulls with no analytic structure are presented. In the third chapter the fact that every rectifiable arc is polynomially convex emerges as a result of the study of the polynomially convex hull of connected sets of finite length. Also in this third chapter, the theory of polynomially convex sets is applied in order to investigate analytic continuation for one dimensional varieties. Further, in the fifth chapter the author studies isoperimetric properties of hulls, results on removable singularities and hulls of surfaces in strictly pseudoconvex boundaries and then, in the sixth chapter he discusses approximation questions both on compact sets and on unbounded sets. The seventh chapter is devoted to the study of one-dimensional subvarieties of strictly pseudoconvex domains. The final chapter discusses some very interesting examples and counterexamples related to topics as polynomial convexity of unions of certain linear spaces, pluripolar graphs, certain deformations…
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32-02
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 32D20
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32E30
zbMATH DE Number: 5115264
zbMATH Keywords: polynomial convexity
zbMATH Keywords: rational convexity
zbMATH Keywords: Stein manifold
zbMATH Keywords: several complex variables
zbMATH Keywords: approximation theory
zbMATH Keywords: polynomially convex hull
zbMATH Keywords: one-dimensional varieties
zbMATH Keywords: analytic structure
zbMATH Keywords: rectifiable arc
zbMATH Keywords: sets of finite length
zbMATH Keywords: removable singularities
zbMATH Keywords: pseudoconvex domains
author: Edgar Lee Stout
reviewed by: Eugen Pascu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5115264