A~criterion for rapid rational approximation in~$\mathbf C^n$

A. S. Sadullaev

Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 271-281

Résumé

This article gives a necessary and sufficient condition for a function which is holomorphic in a neighborhood of zero to belong to the class $R^0$. 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 $R^0$). In particular, Theorem. If $\mathscr O(D)\subset R^0$, then the complement $\mathbf C^n\setminus\widehat D$ of the envelope of holomorphy $D$ 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 $\mathscr O(D)\subset R^0$: this property is satisfied if and only if $\mathbf C^n\setminus\widehat D$ is a pluripolar set. Bibliography: 11 titles.