Polynomial convexity
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 of convex sets and sets with symmetries. The book is not self contained, as the topics discussed cover a large area of the domain. Several basic results of the theory of several complex variables and on Stein manifolds, on functional analysis and classical functions theory are supposed known, but very accurate references are given for results in algebraic topology, Morse theory or geometric measure theory and for other technical results. A comprehensive list of 379 references is given. The style is rigorous, elegant and clear, the exposition is beautiful. The book is an extremely important tool to every researcher interested in the subject, as it contains basic facts and therefore will remain a standard reference in the future and, moreover, it opens a perspective on further directions of research.
scientific article; zbMATH DE number 5115264