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The tame and the wild automorphisms of polynomial rings in three variables

Authors: Ivan P. Shestakov and Ualbai U. Umirbaev
Journal: J. Amer. Math. Soc. 17 (2004), 197-227
MSC (2000): Primary 13F20, 13P10, 14H37; Secondary 14R10, 14R15
Published electronically: October 3, 2003
MathSciNet review: 2015334
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Abstract: A characterization of tame automorphisms of the algebra $A=F[x_1,x_2,x_3]$ of polynomials in three variables over a field $F$ of characteristic $0$ is obtained. In particular, it is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of $A$ are algorithmically recognizable.

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  • 1. A. T. Abdykhalykov, A. A. Mikhalev, U. U. Umirbaev, Automorphisms of two generated free Leibniz algebras, Commun. Algebra. 29 (2001), 2953-2960. MR 2002e:17004
  • 2. H. Bass, Automorphisms of Polynomial Rings, Lecture Notes in Math., 1006, Springer-Verlag, Berlin, pp. 762-771. MR 85b:13009
  • 3. H. Bass, A non-triangular action of $G_a$ on $A^3$, J. of Pure and Appl. Algebra 33 (1984), no. 1, 1-5.
  • 4. G. M. Bergman, Wild automorphisms of free P.I. algebras and some new identities, preprint.
  • 5. P. M. Cohn, Free rings and their relations, 2nd edition, Academic Press, London, 1985. MR 87e:16006
  • 6. A. G. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2, I, II, Trans. Amer. Math. Soc. 160 (1971), 393-401; 171 (1972), 309-315. MR 43:6269; MR 46:9124
  • 7. V. Drensky, J.-T. Yu, Tame and wild coordinates of $K[z][x,y]$, Trans. Amer. Math. Soc. 353 (2001), no. 2, 519-537. MR 2001f:13028
  • 8. A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser-Verlag, Basel, 2000. MR 2001j:14082
  • 9. H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. reine angew. Math. 184(1942), 161-174. MR 5:74f
  • 10. W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde. (3) 1 (1953), 33-41. MR 14:941f
  • 11. L. Makar-Limanov, The automorphisms of the free algebra of two generators, Funksional. Anal. i Prilozhen. 4 (1970), no. 3, 107-108. MR 42:6044
  • 12. M. Nagata, On the automorphism group of $k[x,y]$, Lectures in Math., Kyoto Univ., Kinokuniya, Tokyo, 1972. MR 49:2731
  • 13. G. A. Noskov, The cancellation problem for a ring of polynomials, Sibirsk. Mat. Zh. 19 (1978), no. 6, 1413-1414. MR 81g:13005
  • 14. D. Shannon, M. Sweedler, Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence, J. Symbolic Comput. 6 (1988), 267-273. MR 90e:13002
  • 15. I. P. Shestakov, Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras, Algebra i logika, 32 (1993), no. 5, 571-584; English translation: in Algebra and Logic, 32 (1993), no. 5, 309-317. MR 95c:17034
  • 16. I. P. Shestakov, U. U. Umirbaev, Poisson brackets and two generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004).
  • 17. M. K. Smith, Stably tame automorphisms, J. Pure and Appl. Algebra 58 (1989), 209-212. MR 90f:13005
  • 18. U. U. Umirbaev, Universal derivations and subalgebras of free algebras, In Proc. 3rd Internat. Conf. in Algebra (Krasnoyarsk, 1993). Walter de Gruyter, Berlin, 1996, 255-271. MR 97c:16030
  • 19. D. Wright, Algebras which resemble symmetric algebras, Ph.D. Thesis, Columbia Univ., New York, 1975.

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Additional Information

Ivan P. Shestakov
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo - SP, 05311–970, Brazil; Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

Ualbai U. Umirbaev
Affiliation: Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan

Keywords: Rings of polynomials, automorphisms, subalgebras
Received by editor(s): January 8, 2003
Published electronically: October 3, 2003
Additional Notes: The first author was supported by CNPq.
The second author was supported by the FAPESP Proc.00/06832-8.
Article copyright: © Copyright 2003 American Mathematical Society

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