A CRITERION FOR RAPID RATIONAL APPROXIMATION IN $ \mathbf{C}^n$

Abstract

This article gives a necessary and sufficient condition for a function which is holomorphic in a neighborhood of zero to belong to the class . This criterion, which is formulated in terms of the Taylor coefficients of the function, is then applied to give a description of the singular set of holomorphic functions of several variables which admit rapid rational approximation relative to Lebesgue measure (i.e., which belongs to the class ). In particular,Theorem. If , then the complement of the envelope of holomorphy is a pluripolar set.This theorem together with a well-known result of A. A. Gonchar gives a complete description of the domains for which : this property is satisfied if and only if is a pluripolar set.Bibliography: 11 titles.