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
DOI:
https://doi.org/10.1090/S0894-0347-03-00440-5
Published electronically:
October 3, 2003
MathSciNet review:
2015334
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Abstract | References | Similar Articles | Additional Information
Abstract: A characterization of tame automorphisms of the algebra of polynomials in three variables over a field
of characteristic
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
are algorithmically recognizable.
- 1. A. T. Abdykhalykov, A. A. Mikhalev, and U. U. Umirbaev, Automorphisms of two-generated free Leibniz algebras, Comm. Algebra 29 (2001), no. 7, 2953–2960. MR 1849114, https://doi.org/10.1081/AGB-100104998
- 2. Hyman Bass, Automorphisms of polynomial rings, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp. 762–771. MR 722665, https://doi.org/10.1007/BFb0103749
- 3.
H. Bass, A non-triangular action of
on
, 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 ed., London Mathematical Society Monographs, vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. MR 800091
- 6. Anastasia J. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2. I, Trans. Amer. Math. Soc. 160 (1971), 393–401. MR 280549, https://doi.org/10.1090/S0002-9947-1971-0280549-1
- 7. Vesselin Drensky and Jie-Tai Yu, Tame and wild coordinates of 𝐾[𝑧][𝑥,𝑦], Trans. Amer. Math. Soc. 353 (2001), no. 2, 519–537. MR 1709773, https://doi.org/10.1090/S0002-9947-00-02617-9
- 8. Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619
- 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. G. Makar-Limanov, The automorphisms of the free algebra with two generators, Funkcional. Anal. i Priložen. 4 (1970), no. 3, 107–108 (Russian). MR 0271161
- 12. Masayoshi Nagata, On automorphism group of 𝑘[𝑥,𝑦], Kinokuniya Book-Store Co., Ltd., Tokyo, 1972. Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 5. MR 0337962
- 13. G. A. Noskov, The cancellation problem for a ring of polynomials, Sibirsk. Mat. Zh. 19 (1978), no. 6, 1413–1414, 1440 (Russian). MR 515197
- 14. David Shannon and Moss Sweedler, Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence, J. Symbolic Comput. 6 (1988), no. 2-3, 267–273. Computational aspects of commutative algebra. MR 988417, https://doi.org/10.1016/S0747-7171(88)80047-6
- 15. I. P. Shestakov, Quantization of Poisson superalgebras and the specialty of Jordan superalgebras of Poisson type, Algebra i Logika 32 (1993), no. 5, 571–584, 587 (1994) (Russian, with Russian summary); English transl., Algebra and Logic 32 (1993), no. 5, 309–317 (1994). MR 1287006, https://doi.org/10.1007/BF02261711
- 16. I. P. Shestakov, U. U. Umirbaev, Poisson brackets and two generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004).
- 17. Martha K. Smith, Stably tame automorphisms, J. Pure Appl. Algebra 58 (1989), no. 2, 209–212. MR 1001475, https://doi.org/10.1016/0022-4049(89)90158-8
- 18. Ualbaĭ U. Umirbaev, Universal derivations and subalgebras of free algebras, Algebra (Krasnoyarsk, 1993) de Gruyter, Berlin, 1996, pp. 255–271. MR 1399590
- 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
Email:
shestak@ime.usp.br
Ualbai U. Umirbaev
Affiliation:
Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan
Email:
umirbaev@yahoo.com
DOI:
https://doi.org/10.1090/S0894-0347-03-00440-5
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