Polynomial Convexity

This comprehensive monograph is devoted to the study of polynomially convex sets, which play an important role in the theory of functions of several complex variables.

Important features of Polynomial Convexity:

*Presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets.

*Motivates the theory with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

*Examines in considerable detail questions of uniform approximation, especially on totally real sets, for the most part on compact sets but with some attention to questions of global approximation on noncompact sets.

*Discusses important applications, e.g., to the study of analytic varieties and to the theory of removable singularities for CR functions.

*Requires of the reader a solid background in real and complex analysis together with some previous experience with the theory of functions of several complex variables as well as the elements of functional analysis.

This beautiful exposition of a rich and complex theory, which contains much material not available in other texts, is destined to be the standard reference for many years, and will appeal to all those with an interest in multivariate complex analysis.

Résumé
This book is devoted to an exposition of the theory of polynomially convex sets.Acompact N subset of C is polynomially convex if it is de?ned by a family, ?nite or in?nite, of polynomial inequalities. These sets play an important role in the theory of functions of several complex variables, especially in questions concerning approximation. On the one hand, the present volume is a study of polynomial convexity per se, on the other, it studies the application of polynomial convexity to other parts of complex analysis, especially to approximation theory and the theory of varieties. N Not every compact subset of C is polynomially convex, but associated with an arbitrary compact set, say X, is its polynomially convex hull, X, which is the intersection of all polynomially convex sets that contain X. Of paramount importance in the study of polynomial convexity is the study of the complementary set X \ X. The only obvious reason for this set to be nonempty is for it to have some kind of analytic structure, and initially one wonders whether this set always has complex structure in some sense. It is not long before one is disabused of this naive hope; a natural problem then is that of giving conditions under which the complementary set does have complex structure. In a natural class of one-dimensional examples, such analytic structure is found. The study of this class of examples is one of the major directions of the work at hand.

Contenu
Some General Properties of Polynomially Convex Sets.- Sets of Finite Length.- Sets of Class A1.- Further Results.- Approximation.- Varieties in Strictly Pseudoconvex Domains.- Examples and Counterexamples.