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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

The hypersurface $ x + x^2y + z^2 + t^3 = 0$ over a field of arbitrary characteristic


Author: Anthony J. Crachiola
Journal: Proc. Amer. Math. Soc. 134 (2006), 1289-1298
MSC (2000): Primary 13A50; Secondary 14J30, 14R20
Posted: October 18, 2005
MathSciNet review: 2199171
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Abstract | References | Similar Articles | Additional Information

Abstract: We develop techniques for computing the AK invariant of domains with arbitrary characteristic. As an example, we show that for any field $ \mathbf{k}$ the ring $ \mathbf{k}[X,Y,Z,T] / (X + X^2 Y + Z^2 + T^3)$ is not isomorphic to a polynomial ring over $ \mathbf{k}$.

References

  • [BM] T. Bandman and L. Makar-Limanov, Affine surfaces with $ AK(S) = \mathbf{C}$, Michigan Math. J. 49(2001), 567-582. MR 1872757 (2003a:14094)
  • [C] A. Crachiola, On the AK invariant of certain domains, Ph.D. thesis, Wayne State University, 2004.
  • [CM] A. Crachiola and L. Makar-Limanov, On the rigidity of small domains, J. Algebra 284 (2005), no. 1, 1-12. MR 2115001
  • [D] H. Derksen, More on the hypersurface $ x+x^2y+z^2+t^3=0$ in $ \mathbf{C}^4$, preprint, 1995, 4 pages.
  • [DHM] H. Derksen, O. Hadas, and L. Makar-Limanov, Newton polytopes of invariants of additive group actions, J. Pure Appl. Algebra 156(2001), 187-197. MR 1808822 (2002g:14068)
  • [Du] A. Dubouloz, Generalized Danielewski surfaces, preprint, 2004, 24 pages.
  • [E] A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progr. Math., Vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619 (2001j:14082)
  • [F] T. Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55(1979), 106-110. MR 0531454 (80j:14029)
  • [HS] H. Hasse and F.K. Schmidt, Noch eine Bergründung der Theorie der höheren Differentialquotienten in einem algebraischen Functionenkörper einer Unbestimmten, J. Reine Angew. Math. 177(1937), 215-237.
  • [KKMR] S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell, $ \mathbf{C}^*$-actions on $ \mathbf{C}^3$ are linearizable, Electron. Res. Announc. Amer. Math. Soc. 3(1997), 63-71. MR 1464577 (98i:14046)
  • [KR] M. Koras and P. Russell, Contractible threefolds and $ \mathbf{C}^*$-actions on $ \mathbf{C}^3$, J. Algebraic Geometry 6(1997), no. 4, 671-695. MR 1487230 (99e:14057)
  • [M1] L. Makar-Limanov, On the hypersurface $ x + x^2 y + z^2 + t^3 =0$ in $ \mathbf{C}^4$, or a $ \mathbf{C}^3$-like threefold which is not $ \mathbf{C}^3$, Israel J. Math. 96(1996), 419-429. MR 1433698 (98a:14052)
  • [M2] L. Makar-Limanov, Again $ x + x^2y + z^2 + t^3 = 0$, Contemp. Math., 369, Amer. Math. Soc., 2005. MR 2126661
  • [Mi] M. Miyanishi, $ G_a$-action of the affine plane, Nagoya Math. J. 41(1971), 97-100. MR 0281719 (43:7434)
  • [MS] M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ. 20(1980), 11-42. MR 0564667 (81h:14020)
  • [R] R. Rentschler, Opérations du groupe additif sur le plane affine, C.R. Acad. Sci. Paris 267(1968), 384-387. MR 0232770 (38:1093)
  • [Ru] P. Russell, On affine-ruled rational surfaces, Math. Ann. 255(1981), 287-302. MR 0615851 (82h:14024)

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Additional Information

Anthony J. Crachiola
Affiliation: Department of Mathematics and Computer Science, Loyola University, New Orleans, Louisiana 70118
Address at time of publication: Department of Mathematical Sciences, Saginaw Valley State University, 7400 Bay Road, University Center, Michigan 48710-0001
Email: crachiola@member.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08171-2
PII: S 0002-9939(05)08171-2
Keywords: AK invariant, additive group action, locally finite iterative higher derivation
Received by editor(s): August 25, 2004
Received by editor(s) in revised form: December 26, 2004
Posted: October 18, 2005
Dedicated: To Professor Leonid Makar-Limanov on the occasion of his sixtieth birthday
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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