no. 6
Miyanishi's characterization of the affine 3-space does not hold in higher dimensions
Annales de l'Institut Fourier, Tome 50 (2000) no. 6, pp. 1649-1669

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

Nous présentons un exemple qui confirme l’assertion du titre.

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Kaliman, Shulim; Zaidenberg, Mikhail. Miyanishi's characterization of the affine 3-space does not hold in higher dimensions. Annales de l'Institut Fourier, Tome 50 (2000) no. 6, pp. 1649-1669. doi: 10.5802/aif.1803

[Ab] Shreeram S. Abhyankar, Quasirational singularities, Amer. J. Math., 101-2 (1979), 267-300. | Zbl | MR

[BaDw] F. Baldassarri, B. Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math., 101 (1979), 42-76. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[Beu] F. Beukers, The Diophantine equation Axp + Byq = Czr, Duke Math. J., 91-1 (1998), 61-88. | Zbl | MR

[Bu] A. Buium, The abc theorem for abelian varieties, Intern. Math. Res. Notices, 5 (1994), 219-232. | Zbl | MR

[BoMu] E. Bombieri, J. Mueller, Trinomial equations in function fields, Astérisque, 22 (1995), 19-40. | Zbl | MR

[Br] B. Brindza, On the equation F(x,y) = zm over function fields, Acta Math. Hung., 49 (1987), 267-275. | Zbl | MR

[ClGr] C. H. Clemens, Ph. A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math., 95 (1972), 281-356. | Zbl | MR

[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. | Zbl

[DarGr] H. Darmon, A. Granville, On the equations zm = F(x,y) and Axp + Byq = Czr, Bull. London Math. Soc., 27 (1995), 513-543. | Zbl

[Dav] H. Davenport, On f3(t) - g2(t), Norske Vid. Selsk. Forh. (Trondheim), 38 (1965), 86-87. | Zbl

[De] H. Derksen, Constructive Invariant Theory and the Linearization Problem, Ph. D. thesis, Basel (1997), 39p. | Zbl | MR

[tDP] T. Tom Dieck, T. Petrie, Contractible affine surfaces of Kodaira dimension one, Japan J. Math., 16 (1990), 147-169. | Zbl | MR

[DvZa] R. Dvornicich, U. Zannier, A note on Thue's equation over function fields, Monatsh. für Math., 118 (1994), 219-230. | Zbl | MR

[Ev] A. Evyatar (formely A. Gutwirth), On polynomial equations, Israel J. Math., 10 (1971), 321-326. | Zbl

[FIZa] H. Flenner, M. Zaidenberg, Rational curves and rational singularities, in preparation.

[Fu] T. Fujita, On the topology of non complete algebraic surfaces, J. Fac. Sci. Univ. Tokyo, Sect. IA, 29 (1982), 503-566. | Zbl | MR

[Grm] M. Gromov, Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897. | Zbl | MR

[Grs] F. Gross, On the functional equation fn + gn = hn, Amer. Math. Monthly, 73 (1966), 1093-1096. | Zbl | MR

[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.

[Ja] A. V. Jategaonkar, Elementary proof of a theorem of P. Montel on entire functions, J. Lond. Math. Soc., 40 (1965), 166-170. | Zbl | MR

[Ka1] S. Kaliman, Exotic analytic structures and Eisenman intrinsic measures, Israel Math. J., 88 (1994), 411-423. | Zbl | MR

[Ka2] S. Kaliman, Polynomials with general ℂ2-fibers are variables.I, preprint (1999), 60p.

[KaML1] S. Kaliman, L. Makar-Limanov, On Russell-Koras contractible threefolds, J. of Algebraic Geom., 6 (1997), 247-268. | Zbl | MR

[KaML2] S. Kaliman, L. Makar-Limanov, Affine algebraic manifolds without dominant morphisms from Euclidean spaces, Rocky Mount. J. Math., 27-2 (1997), 601-609. | Zbl | MR

[KaML3] S. Kaliman, L. Makar-Limanov, Locally nilpotent derivations of Jacobian type, preprint (1998), 16p.

[KaZa1] S. Kaliman, M. Zaidenberg, Affine modifications and affine varieties with a very transitive automorphism group, Transformation Groups, 4-1 (1999), 53-95. | Zbl | MR

[KaZa2] S. Kaliman, M. Zaidenberg, Families of affine planes: the existence of a cylinder, MPI, preprint MPI 00-75 (2000), 12p.

[KlNe] M. C. Klamkin, D. J. Newman, On the number of distinct zeros of polynomials, Amer. Mathem. Monthly, 66 (1959), 494-496. | Zbl | MR

[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.

[La] S. Lang, Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. | Zbl | MR

[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. | Zbl | MR

[ML2] L. Makar-Limanov, On groups of automorphisms of a class of surfaces, Israel J. Math., 69 (1990), 250-256. | Zbl | MR

[ML3] L. Makar-Limanov, On the group of automorphisms of a surface xny = P(z), Preprint (1997), 11p.

[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). | Zbl

[Mas] R. C. Mason, Equations over function fields, Number theory, Noordwijkerhout 1983. Lecture Notes in Math., 1068, Springer, Berlin-New York (1984), 149-157. | Zbl | MR

[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. | Zbl | MR

[Miy] M. Miyanishi, Algebraic characterization of the affine 3-space, Proc. Algebraic Geom. Seminar, Singapore, World Scientific (1987), 53-67.

[Ore1] S. Yu. Orevkov, Riemann existence theorem and construction of real algebraic curves, preprint (1999), 1-11.

[Ore2] S. Yu. Orevkov, On singularities that are quasirational in the sense of Abhyankar, Uspekhi Mat. Nauk, 50 (1995), no. 6 (306), 201-202. | Zbl | MR

[OrlWa] P. Orlik, P. Wagreich, Algebraic surfaces with k*-action, Acta Math., 138 (1977), 43-81. | Zbl | MR

[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. | Zbl | MR

[Pr] V. V. Prasolov, Mnogochleny (Polynomials) (in Russian), MCNMO, Moscow, 2000.

[Sa] A. Sathaye, Polynomial ring in two variables over a D. V. R.: A criterion, Invent. Math., 74 (1983), 159-168. | Zbl | MR

[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. | Zbl

[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. | JFM

[Si] J. H. Silverman, The S-unit equation over function fields, Math. Proc. Cambridge Philos. Soc., 95 (1984), 3-4. | Zbl | MR

[St] W. W. Stothers, Polynomial identities and Hauptmoduln, Quart. J. Math. (2), 32 (1981), 349-370. | Zbl | MR

[Ve] J. Végsö, On power values of binary forms over function fields, Publ. Math. Debrecen, 50-1, 2 (1997), 145-148. | Zbl | MR

[Wi] J. Winkelmann, On automorphisms of complements of analytic subsets in Cn, Math. Zeitschrift, 204 (1990), 117-127. | Zbl | MR

[Za1] M. Zaidenberg, An analytic cancellation theorem and exotic algebraic structures on ℂn, n ≥ 3, Astérisque, 217 (1993), 251-282. | Zbl

[Za2] M. Zaidenberg, On exotic algebraic structures on affine spaces, Algebra and Analysis. St. Petersbourg Mathem. J., 10-5 (1999), 60p.

[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. | Zbl | MR

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