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The first chapter is introductory. It presents some of the basic notions of our subject, it assembles some of the tools and techniques that will be used throughout the text, and it presents some examples. Section 1.1 introduces the notion of polynomial convexity and the related notion of rational convexity. Section 1.2 is an introduction to the abstract theory of uniform algebras. Section 1.3 summarizes some parts of the theory of plurisubharmonic functions. Section 1.4 is devoted to the Cauchy-Fantappiè integral, a very general integral formula in the setting of the theory of functions of several complex variables, whether on ℂ or on complex manifolds. Section 1.5 contains a proof of the Oka-Weil approximation theorem based on the Cauchy-Fantappiè formula. Section 1.6 presents several examples, some of which indicate the pathology of the subject, others of which are results that will find application in the sequel. Section 1.7 gives an example of a hull with no analytic structure.

The main properties of polynomially convex sets discussed in this chapter are of a general character in that they do not depend on particular structural properties of the sets involved. Section 2.1 contains some of the information about polynomially convex sets that can be derived from the theory of the Cousin problems. Section 2.2 contains two characterizations of polynomially convex sets. Section 2.3 brings the geometric methods of Morse theory and algebraic topology to bear on polynomial convexity. Section 2.4 is devoted to some results for various classes of compacta in Stein manifolds that are parallel to results for polynomially convex subsets of ℂ.

This chapter is devoted to a fairly self-contained discussion of the polynomially convex hull of a connected set of finite length or, more generally, a set that is contained in a connected set of finite length. One result that finally emerges in this chapter is that each rectifiable arc in ℂ is polynomially convex. Much of the chapter is devoted to preliminaries from classical function theory and from real analysis. Section 3.1 contains a statement of one of the principal results of the chapter and some remarks about it. Section 3.2 assembles well-known information about one-dimensional analytic varieties. Section 3.3 contains geometric preliminaries concerning Hausdorff measures, integration, and sets of finite length. Section 3.4 is devoted to some essential results on conformal mapping and related issues. Section 3.5 establishes the subharmonicity of certain functions naturally associated with the polynomially convex hull of a compact set. Section 3.6 shows that the polynomially convex hull of a connected set of finite length is a one-dimensional variety. Section 3.7 shows that this hull has finite area. Section 3.8 applies the preceding theory to the continuation of one-dimensional varieties and, vice versa, this theory of continuation to the study of hulls.

In this chapter we discuss the hulls of a class of sets with finite length more general than those contained in connected sets. Section 4.1 is introductory. Section 4.2 assembles some results from geometric measure theory for use in subsequent sections. Section 4.3 introduces the class of sets that are the main object of study in this chapter. Section 4.4 establishes the finiteness of the area of certain one-dimensional varieties. Section 4.5 contains a version of Stokes’s theorem. Section 4.6 introduces a useful multiplicity function. Section 4.7 contains a bound on the number of global branches of a hull.

This chapter treats some further results in the theory of polynomial convexity, which are rather loosely related but draw substantially on the work of the preceding chapters. Section 5.1 discusses isoperimetric questions in the context of polynomial convexity. Section 5.2 considers some questions in the theory of removable singularities for holomorphic functions and their boundary values. Section 5.3 treats certain convexity problems for two-dimensional surfaces in three-dimensional strictly pseudoconvex boundaries.

The present chapter is devoted in the main to some approximation theorems for continuous functions defined on totally real sets. The approximation results established here are of a global nature. Section 6.1 contains preparatory material on totally real sets and manifolds. Section 6.2 introduces holomorphically convex compacta and develops some of their main properties. Section 6.3 contains a result on uniform approximation on compacta in totally real sets. Section 6.4 presents some material on the algebras ℛ() for planar compacta for use in the following section. Section 6.5 considers algebras on smooth manifolds and analytic varieties. Section 6.6 contains results on tangential approximation.

This chapter is devoted to the study of one-dimensional subvarieties of strictly pseudoconvex domains. The motivation comes in good measure from the highly developed theory of the boundary behavior of holomorphic functions; the present chapter may be regarded as presenting some results toward an analogous geometric theory for varieties. Section 7.1 contains work on interpolation, which serves as a tool in the subsequent sections. Section 7.2 treats boundary regularity questions. Section 7.3 considers boundary uniqueness results.

In this chapter we discuss some additional questions related to polynomial convexity, topics that are concerned with polynomial convexity per se and also topics that depend on the application of the ideas of polynomial convexity. Section 8.1 discusses the polynomial convexity of unions of linear spaces passing through the origin. Section 8.2 is devoted to the study of pluripolar graphs. Section 8.3 considers certain deformations of polynomially convex sets. Section 8.4 concerns sets with symmetries.