Miyanishi's characterization of the affine 3-space does not hold in higher dimensions

Shulim Kaliman; Mikhail Zaidenberg

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 6, page 1649-1669
  • ISSN: 0373-0956


We present an example which confirms the assertion of the title.

How to cite


Kaliman, Shulim, and Zaidenberg, Mikhail. "Miyanishi's characterization of the affine 3-space does not hold in higher dimensions." Annales de l'institut Fourier 50.6 (2000): 1649-1669. <http://eudml.org/doc/75467>.

abstract = {We present an example which confirms the assertion of the title.},
author = {Kaliman, Shulim, Zaidenberg, Mikhail},
journal = {Annales de l'institut Fourier},
keywords = {affine space; polynomial algebras; exotic structure},
language = {eng},
number = {6},
pages = {1649-1669},
publisher = {Association des Annales de l'Institut Fourier},
title = {Miyanishi's characterization of the affine 3-space does not hold in higher dimensions},
url = {http://eudml.org/doc/75467},
volume = {50},
year = {2000},

AU - Kaliman, Shulim
AU - Zaidenberg, Mikhail
TI - Miyanishi's characterization of the affine 3-space does not hold in higher dimensions
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 6
SP - 1649
EP - 1669
AB - We present an example which confirms the assertion of the title.
LA - eng
KW - affine space; polynomial algebras; exotic structure
UR - http://eudml.org/doc/75467
ER -


  1. [Ab] Shreeram S. ABHYANKAR, Quasirational singularities, Amer. J. Math., 101-2 (1979), 267-300. Zbl0425.14009MR80h:14014
  2. [BaDw] F. BALDASSARRI, B. DWORK, On second order linear differential equations with algebraic solutions, Amer. J. Math., 101 (1979), 42-76. Zbl0425.34007MR81d:34002
  3. [BarKa] G. BARTHEL, L. KAUP, Topologie des surfaces complexes compactes singulières, in: Sur la topologie des surfaces complexes compactes, Sém. Math. Sup., 80, Presses Univ. Montréal, Montréal, Que. (1982), 61-297. Zbl0494.32011MR84j:32035
  4. [Be] J. BERTIN, Automorphismes des surfaces non complètes, groupes fuchsiens et singularités quasihomogènes, Sémin. d'algèbre P. Dubreil et M.-P. Malliavin, 36ème Année, Proc., Paris 1983/84, Lect. Notes Math., 1146 (1985), 106-126. Zbl0577.14024MR88a:14040
  5. [Beu] F. BEUKERS, The Diophantine equation Axp + Byq = Czr, Duke Math. J., 91-1 (1998), 61-88. Zbl1038.11505MR99f:11039
  6. [Bu] A. BUIUM, The abc theorem for abelian varieties, Intern. Math. Res. Notices, 5 (1994), 219-232. Zbl0836.14025MR95c:11074
  7. [BoMu] E. BOMBIERI, J. MUELLER, Trinomial equations in function fields, Astérisque, 22 (1995), 19-40. Zbl0824.11013MR96g:11026
  8. [Br] B. BRINDZA, On the equation F(x,y) = zm over function fields, Acta Math. Hung., 49 (1987), 267-275. Zbl0613.10018MR88d:11028
  9. [ClGr] C. H. CLEMENS, Ph. A. GRIFFITHS, The intermediate Jacobian of the cubic threefold, Ann. of Math., 95 (1972), 281-356. MR46 #1796
  10. [DanGi] V.I. DANILOV, M.H. GIZATULLIN, Automorphisms of affine surfaces. I, II, Math. USSR Izv., 9 (1975), 493-534; ibid., 11 (1977), 51-98. Zbl0379.14002
  11. [DarGr] H. DARMON, A. GRANVILLE, On the equations zm = F(x,y) and Axp + Byq = Czr, Bull. London Math. Soc., 27 (1995), 513-543. Zbl0838.11023
  12. [Dav] H. DAVENPORT, On f3(t) - g2(t), Norske Vid. Selsk. Forh. (Trondheim), 38 (1965), 86-87. Zbl0136.25204
  13. [De] H. DERKSEN, Constructive Invariant Theory and the Linearization Problem, Ph. D. thesis, Basel (1997), 39p. Zbl0883.13003MR98m:13010
  14. [tDP] T. tom DIECK, T. PETRIE, Contractible affine surfaces of Kodaira dimension one, Japan J. Math., 16 (1990), 147-169. Zbl0721.14018MR91j:14027
  15. [DvZa] R. DVORNICICH, U. ZANNIER, A note on Thue's equation over function fields, Monatsh. für Math., 118 (1994), 219-230. Zbl0820.11019MR95j:11028
  16. [Ev] A. EVYATAR (formely A. GUTWIRTH), On polynomial equations, Israel J. Math., 10 (1971), 321-326. Zbl0231.12002
  17. [FIZa] H. FLENNER, M. ZAIDENBERG, Rational curves and rational singularities, in preparation. Zbl1043.14008
  18. [Fu] T. FUJITA, On the topology of non complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo, Sect. IA, 29 (1982), 503-566. Zbl0513.14018MR84g:14035
  19. [Grm] M. GROMOV, Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897. Zbl0686.32012MR90g:32017
  20. [Grs] F. GROSS, On the functional equation fn + gn = hn, Amer. Math. Monthly, 73 (1966), 1093-1096. Zbl0154.40104MR34 #4494
  21. [Ha] G.H. HALPHEN, Sur la réduction des équations différentielles linéaires aux formes intégrables, Mémoires présentés par divers savants à l'Académie des sciences de l'Institut National de France, T. XXVIII, N 1, Paris, F. Krantz, 1883; œuvres. Vol. 3, Paris (1921), 1-260. 
  22. [Ja] A. V. JATEGAONKAR, Elementary proof of a theorem of P. Montel on entire functions, J. Lond. Math. Soc., 40 (1965), 166-170. Zbl0134.05403MR30 #248
  23. [Ka1] S. KALIMAN, Exotic analytic structures and Eisenman intrinsic measures, Israel Math. J., 88 (1994), 411-423. Zbl0821.14025MR95j:32038
  24. [Ka2] S. KALIMAN, Polynomials with general ℂ2-fibers are variables.I, preprint (1999), 60p. 
  25. [KaML1] S. KALIMAN, L. MAKAR-LIMANOV, On Russell-Koras contractible threefolds, J. of Algebraic Geom., 6 (1997), 247-268. Zbl0897.14010MR98m:14041
  26. [KaML2] S. KALIMAN, L. MAKAR-LIMANOV, Affine algebraic manifolds without dominant morphisms from Euclidean spaces, Rocky Mount. J. Math., 27-2 (1997), 601-609. Zbl0951.14007MR98i:14042
  27. [KaML3] S. KALIMAN, L. MAKAR-LIMANOV, Locally nilpotent derivations of Jacobian type, preprint (1998), 16p. 
  28. [KaZa1] S. KALIMAN, M. ZAIDENBERG, Affine modifications and affine varieties with a very transitive automorphism group, Transformation Groups, 4-1 (1999), 53-95. Zbl0956.14041MR2000f:14099
  29. [KaZa2] S. KALIMAN, M. ZAIDENBERG, Families of affine planes: the existence of a cylinder, MPI, preprint MPI 00-75 (2000), 12p. 
  30. [KlNe] M. C. KLAMKIN, D. J. NEWMAN, On the number of distinct zeros of polynomials, Amer. Mathem. Monthly, 66 (1959), 494-496. Zbl0089.01003MR21 #2611
  31. [Kl] F. KLEIN, Vorlesungen über das ikosaeder und die auflösung der gleichungen vom fünden grade, Teubner, Leipzig, 1884. English transl.: F. KLEIN, Lectures on the Icosahedron and the solution of equations of fifth degree, Dover, 1956. JFM16.0061.01
  32. [La] S. LANG, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. Zbl0714.11034MR90k:11032
  33. [ML1] L. MAKAR-LIMANOV, On the hypersurface x + x2y + z2 +t3 = 0 in ℂ4 or a ℂ3-like threefold which is not ℂ3, Israel J. Math., 96 (1996), 419-429. Zbl0896.14021MR98a:14052
  34. [ML2] L. MAKAR-LIMANOV, On groups of automorphisms of a class of surfaces, Israel J. Math., 69 (1990), 250-256. Zbl0711.14022MR91b:14059
  35. [ML3] L. MAKAR-LIMANOV, On the group of automorphisms of a surface xny = P(z), Preprint (1997), 11p. 
  36. [Man] Yu. I. MANIN, Cubic forms. Algebra, geometry, arithmetic, North-Holland Mathematical Library, 4. North-Holland Publishing Co., Amsterdam-New York (1986), 326 p. (The Russian original "Nauka", Moscow, 1972). Zbl0582.14010
  37. [Mas] R. C. MASON, Equations over function fields, Number theory, Noordwijkerhout 1983. Lecture Notes in Math., 1068, Springer, Berlin-New York (1984), 149-157. Zbl0544.10015MR86a:11012
  38. [Mil] J. MILNOR, On the 3-dimensional Brieskorn manifolds M(p, q, r), in: Knots, groups, and 3-manifolds, L. P. Neuwirth, ed. Annals of Math. Stud., Princenton Univ. Press, Princeton, NJ (1975), 175-225. Zbl0305.57003MR54 #6169
  39. [Miy] M. MIYANISHI, Algebraic characterization of the affine 3-space, Proc. Algebraic Geom. Seminar, Singapore, World Scientific (1987), 53-67. 
  40. [Ore1] S. Yu. OREVKOV, Riemann existence theorem and construction of real algebraic curves, preprint (1999), 1-11. 
  41. [Ore2] S. Yu. OREVKOV, On singularities that are quasirational in the sense of Abhyankar, Uspekhi Mat. Nauk, 50 (1995), no. 6 (306), 201-202. Zbl0858.32029MR96k:32077
  42. [OrlWa] P. ORLIK, P. WAGREICH, Algebraic surfaces with k*-action, Acta Math., 138 (1977), 43-81. Zbl0352.14016MR57 #336
  43. [PaVa] G. PAYNE, L. VASERSTEIN, Sums of three cubes, The arithmetic of function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin (1992), 443-454. Zbl0799.11039MR93k:11090
  44. [Pr] V. V. PRASOLOV, Mnogochleny (Polynomials) (in Russian), MCNMO, Moscow, 2000. 
  45. [Sa] A. SATHAYE, Polynomial ring in two variables over a D. V. R.: A criterion, Invent. Math., 74 (1983), 159-168. Zbl0538.13006MR85j:14098
  46. [Sch] W. M. SCHMIDT, Polynomial solutions of F(x,y) = zn, Proceedings of the Queen's Number Theory Conference, 1979 (Kingston, Ont., 1979), Queen's Papers in Pure and Appl. Math., 54, Queen's Univ., Kingston, Ont. (1980), 33-65. Zbl0454.10010
  47. [Schw] H. A. SCHWARTZ, Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. für die reine und angewandte Mathematik, 75 (1873), 292-335. JFM05.0146.03
  48. [Si] J. H. SILVERMAN, The S-unit equation over function fields, Math. Proc. Cambridge Philos. Soc., 95 (1984), 3-4. Zbl0533.10013MR85e:11018
  49. [St] W. W. STOTHERS, Polynomial identities and Hauptmoduln, Quart. J. Math. (2), 32 (1981), 349-370. Zbl0466.12011MR83m:12006
  50. [Ve] J. VÉGSÖ, On power values of binary forms over function fields, Publ. Math. Debrecen, 50-1, 2 (1997), 145-148. Zbl0880.11032MR98a:11039
  51. [Wi] J. WINKELMANN, On automorphisms of complements of analytic subsets in Cn, Math. Zeitschrift, 204 (1990), 117-127. Zbl0701.32014MR91e:32029
  52. [Za1] M. ZAIDENBERG, An analytic cancellation theorem and exotic algebraic structures on ℂn, n ≥ 3, Astérisque, 217 (1993), 251-282. Zbl0801.14001
  53. [Za2] M. ZAIDENBERG, On exotic algebraic structures on affine spaces, Algebra and Analysis. St. Petersbourg Mathem. J., 10-5 (1999), 60p. 
  54. [Zn] U. ZANNIER, On Davenport's bound for the degree of f3 - g2 and Riemann's existence theorem, Acta Arithm., 71 (1995), 107-137; Addenda, ibid., 74 (1996), 387. Zbl0840.11015MR96k:11029a

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.