Polynomial approximation on parabolic manifolds

A. Sadullaev   ; A. A. Atamuratov

Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 703-716

Résumé

On a parabolic manifold polynomials are defined in terms of a special exhaustion function, and the problem of polynomial approximation of analytic functions is considered. An example of a parabolic manifold on which the class of polynomials consists of the constants alone is presented. On regularly parabolic manifolds, which possess a large reserve of polynomials, an analogue of the celebrated Bernstein–Walsh theorem is proved. Bibliography: 28 titles.

Détail

Keywords: plurisubharmonic functions, Stein parabolic manifolds, exhaustion function, polynomials, rapid approximation.

Bibliographie

[1] P. Griffits and J. King, “Nevanlinna theory and holomorphic mappings between algebraic varieties”, Acta Math., 130 (1973), 145–220 | DOI | MR | Zbl

[2] W. Stoll, Value distribution on parabolic spaces, Lecture Notes in Math., 600, Springer-Verlag, Berlin–New York, 1977, viii+216 pp. | DOI | MR | Zbl

[3] W. Stoll, “The characterization of strictly parabolic manifolds”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7:1 (1980), 87–154 | MR | Zbl

[4] A. Sadullaev, “Deficient divisors in the Valiron sense”, Math. USSR-Sb., 36:4 (1980), 535–547 | DOI | MR | Zbl

[5] D. Burns, “Curvatures of Monge–Ampère foliations and parabolic manifolds”, Ann. of Math. (2), 115:2 (1982), 349–373 | DOI | MR | Zbl

[6] J.-P. Demailly, Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France (N.S.), 19, Soc. Math. France, Paris, 1985, 124 pp. | MR | Zbl

[7] A. Zeriahi, “Function de Green pluricomplexe à pole à l'infini sur un espace de Stein parabolique et applications”, Math. Scand., 69:1 (1991), 89–126 | DOI | MR | Zbl

[8] R. L. Foote, “Homogeneous complex Monge–Ampère equations and algebraic embeddings of parabolic manifolds”, Indiana Univ. Math. J., 39:4 (1990), 1245–1273 | DOI | MR | Zbl

[9] Yuancheng Liu and Min Ru, “A defect relation for meromorphic maps on parabolic manifolds intersecting hypersurfaces”, Illinois J. Math., 49:1 (2005), 237–257 | DOI | MR | Zbl

[10] A. Aytuna and A. Sadullaev, “$S^*$-parabolic manifolds”, TWMS J. Pure Appl. Math., 2:1 (2011), 6–9 | MR | Zbl

[11] Qi Han, “A defect relation for meromorphic maps on generalized $p$-parabolic manifolds intersecting hypersurfaces in complex projective algebraic varieties”, Proc. Edinb. Math. Soc. (2), 56:2 (2013), 551–574 | DOI | MR | Zbl

[12] M. Kalka and G. Patrizio, “Splitting parabolic manifolds”, Riv. Math. Univ. Parma (N.S.), 5:2 (2014), 443–453 | MR | Zbl

[13] A. Aytuna and A. Sadullaev, “Parabolic Stein manifolds”, Math. Scand., 114:1 (2014), 86–109 | DOI | MR | Zbl

[14] A. Aytuna and A. Sadullaev, “Polynomials on parabolic manifolds”, Topics in several complex variables, Contemp. Math., 662, Amer. Math. Soc., Providence, RI, 2016, 1–22 | DOI | MR | Zbl

[15] Wei Chen and Nguyen Van Thin, “A general form of Second Main Theorem for meromorphic mappings from a $p$-parabolic manifold to a projective algebraic variety”, Indian J. Pure Appl. Math., 52:3 (2021), 847–860 | DOI | MR | Zbl

[16] A. A. Atamuratov, “Polynomials on regular parabolic manifolds”, J. Math. Sci. (N.Y.), 278:4 (2024), 596–612 | DOI | MR | Zbl

[17] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Grundlehren Math. Wiss., 164, Springer-Verlag, New York–Berlin, 1970, xx+446 pp. | DOI | MR | Zbl

[18] H. Federer, Geometric measure theory, Grundlehren Math. Wiss., 153, Springer-Verlag, New York, 1969, xiv+676 pp. | DOI | MR | Zbl

[19] W. Rudin, “A geometric criterion for algebraic varieties”, J. Math. Mech., 17:7 (1968), 671–683 | MR | Zbl

[20] A. Sadullaev, “Criteria for the algebraicity of analytic sets”, Holomorphic functions of several complex variables, Institute of Physics of the Siberian Branch of the USSR Academy of Sciences, Krasnoyarsk, 1976, 107–122 (Russian)

[21] J. Siciak, “On some extremal functions and their applications in the theory of analytic functions of several complex variables”, Trans. Amer. Math. Soc., 105:2 (1962), 322–357 | DOI | MR | Zbl

[22] A. A. Gončar (Gonchar), “A local condition for the single-valuedness of analytic functions of several variables”, Math. USSR-Sb., 22:2 (1974), 305–322 | DOI | MR | Zbl

[23] E. M. Čirka (Chirka), “Meromorphic continuation and the degree of rational approximations in $\mathbf C^N$”, Math. USSR-Sb., 28:4 (1976), 553–561 | DOI | MR | Zbl

[24] V. P. Zakharyuta, “Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein–Walsh theorem for analytic functions of several complex variables”, Ann. Polon. Math., 33:1–2 (1976), 137–148 (Russian) | DOI | MR | Zbl

[25] A. Sadullaev, “Rational approximation and pluripolar sets”, Math. USSR-Sb., 47:1 (1984), 91–113 | DOI | MR | Zbl

[26] A. Sadullaev, “A criterion for rapid rational approximation in $\mathbf C^n$”, Math. USSR-Sb., 53:1 (1986), 271–281 | DOI | MR | Zbl

[27] A. Sadullaev, “Plurisubharmonic functions”, Several complex variables. II. Function theory in classical domains. Complex potential theory, Encyclopaedia Math. Sci., 8, Springer, Berlin, 1994, 59–106 | MR | Zbl

[28] F. Forstnerič, Stein manifolds and holomorphic mappings. The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. (3), 56, 2nd ed., Springer, Cham, 2017, xiv+562 pp. | DOI | MR | Zbl