Polynomials on regular parabolic manifolds

A. A. Atamuratov

Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 41-58.

Résumé

In this work, we consider the regular parabolic manifold $X$ and polynomials on it. We prove some properties of regular parabolic manifolds and describe polynomials on complements of Weierstrass algebroidal sets.

Bibliographie

[1] Danchenko V. I., “Dliny lemniskat. Variatsii ratsionalnykh funktsii”, Mat. sb., 198:8 (2007), 1111–1117 | Zbl

[2] Sadullaev A. S., “Defektnye divizory v smysle Valirona”, Mat. sb., 108:4 (1979), 567–580 | MR

[3] Sadullaev A. S., Chirka E. M., “O prodolzhenii funktsii s polyarnymi osobennostyami”, Mat. sb., 132:3 (1987), 383–390 | Zbl

[4] Aytuna A., Krone J., Terzioglu T., “Complemented infinite type power series subspaces of nuclear Frechet spaces”, Math. Ann., 283:2 (1989), 193–202 | DOI | MR | Zbl

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

[6] Aytuna A., Sadullaev A., “Polynomials on parabolic manifolds”, Topics in several complex variables, First USA–Uzbekistan conference on analysis and mathematical physics (California State University, Fullerton, CA, USA, May 20–23, 2014), Am. Math. Soc., Providence, 2016, 1–22 | MR | Zbl

[7] Bedford E., Kalka M., “Foliations and complex Monge—Ampere equations”, Commun. Pure Appl. Math., 30 (1977), 543–571 | DOI | MR | Zbl

[8] Demailly J. P., “Mesures de Monge—Ampere et caractrisation geometrique des variets algebriques”, Mem. Soc. Math. Fr. (N.S.), 19 (1985), 1–124 | MR

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

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

[11] Kalka M., Patrizio G., Splitting parabolic manifolds, 2014, arXiv: 1409.3972v1 [math.CV] | MR

[12] Pomerenko Ch., “On derivative of a polynomial”, Michigan Math. J., 6:4 (1959), 373–375 | MR

[13] Poppenberg M., “Tame subspaces of power series spaces”, Functional analysis, Proceedings of the first international workshop (Trier University, Germany, September 26–October 1, 1994), de Gruyter, Berlin, 1996, 365–375 | MR | Zbl

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

[15] Shabat B. V., Introduction to complex analysis, v. II, Functions of several variables, Am. Math. Soc., Providence, 1992 | MR | Zbl

[16] Snaebjarnarson A. S., Rapid polynomial approximation on Stein manifolds, 2016, arXiv: 1612.06173v1 [math.CV] | MR

[17] Stoll W., Value distribution on parabolic spaces, Springer, Berlin—Heidelberg—New York, 1977 | MR | Zbl

[18] Stoll W., “The characterization of strictly parabolic manifolds”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 7 (1980), 87–154 | MR | Zbl

[19] Vogt D., “Charakterisierung der Unterräume von (s)”, Math. Z., 155:2 (1977), 109–111 | DOI | MR

[20] Zeriahi A., “Function de Green pluricomplex a pole a l'infini sur un espace de Stein parabolique”, Math. Scand., 69 (1991), 89–126 | DOI | MR | Zbl