Polynomial Convexity [Hardcover]

This comprehensive monograph details polynomially convex sets. It presents the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets. Coverage examines in considerable detail questions of uniform approximation for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. The book also discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

This comprehensive monograph details polynomially convex sets. It discusses important applications and motivates the reader with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries.

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 isl¢