Geodesic
A criterion for rapid rational approximation in $\mathbf C^n$
Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 271-281 
A. S. Sadullaev. A criterion for rapid rational approximation in $\mathbf C^n$. Sbornik. Mathematics, Tome 53 (1986) no. 1, pp. 271-281. http://geodesic.mathdoc.fr/item/SM_1986_53_1_a13/
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     title = {A~criterion for rapid rational approximation in~$\mathbf C^n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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[2] Gonchar A. A., “O skhodimosti approksimatsii Pade”, Matem. sb., 92(134) (1973), 152–164 | Zbl

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[6] Sadullaev A., “Ratsionalnye approksimatsii i plyuripolyarnye mnozhestva”, Matem. sb., 119(161) (1982), 96–118 | MR | Zbl

[7] Gonchar A. A., “O ravnomernoi skhodimosti diagonalnykh approksimatsii Pade”, Matem. sb., 118(160) (1982), 535–556 | MR | Zbl

[8] Alexander H., “Prosective capacity”, Ann. Math. Stud., 100 (1981), 3–27 | MR | Zbl

[9] Oka K., “Notes sur les families des fonctions analytiques multiform etc”, J. Sci. Hiroshima Univ., ser. A, 4 (1934), 93–98 | Zbl

[10] Nishino T., “Sur les ensembles pseudo-concaves”, J. Math. Kyoto Univ., I (1962), 225–245 | MR

[11] Slodkowski Z., “Analytic set-valued functions and spectra”, Math. Annalen, 256(3) (1981), 363–386 | DOI | MR | Zbl