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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Tame and Wild Coordinates of $K[z][x,y]$

Author(s): Vesselin Drensky; Jie-Tai Yu
Journal: Trans. Amer. Math. Soc. 353 (2001), 519-537.
MSC (2000): Primary 13B25; Secondary 13B10, 13P10, 14E07
Posted: October 19, 2000
MathSciNet review: 1709773
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Abstract:

Let $K$ be a field of characteristic zero. We characterize coordinates and tame coordinates in $K[z][x,y]$, i.e. the images of $x$ respectively under all automorphisms and under the tame automorphisms of $K[z][x,y]$. We also construct a new large class of wild automorphisms of $K[z][x,y]$ which maps $x$ to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of $K[z][x,y,t]$.

References:

1.
J. Alev, A note on Nagata's automorphism, in A. van den Essen (ed.), Automorphisms of Affine Spaces, Kluwer Acad. Publ., Dordrecht, 1995, 215-221. MR 97d:14023

2.
W.W. Adams, P. Loustaunau, An Introduction to Gröbner Bases, Graduate Studies in Math. 3, AMS, Providence, R.I., 1994. MR 95g:13025

3.
H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330. MR 83k:14028

4.
C. Cheng, S. Wang, An algorithm that determines whether a polynomial map is bijective, in A. van den Essen (ed.), Automorphisms of Affine Spaces, Kluwer Acad. Publ., Dordrecht, 1995, 169-176. MR 96h:14021

5.
P.M. Cohn, Free Rings and Their Relations, Second Edition, Acad. Press, 1985. MR 87e:16006

6.
D. Daigle, G. Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of $k[X_{1},\ldots ,X_{n}]$, J. Algebra 204 (1998), 353-371. MR 99c:13011

7.
V. Drensky, A. van den Essen, D. Stefanov, New stably tame automorphisms of polynomial algebras, J. Algebra 226 (2000), 629-638.

8.
V. Drensky, J. Gutierrez, J.-T. Yu, Gröbner bases and the Nagata automorphism, J. Pure Appl. Algebra 135 (1999), 135-153. MR 2000b:13011

9.
A. van den Essen, A criterion to decide if a polynomial map is invertible and to compute the inverse, Commun. Algebra 18 (1990), 3183-3186. MR 91e:13023

10.
H.W.E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine und Angew. Math. 184 (1942), 161-174. MR 5:74f

11.
O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.

12.
W. van der Kulk, On polynomial rings in two variables, Nieuw Archief voor Wiskunde (3) 1 (1953), 33-41. MR 14:941f

13.
L. Le Bruyn, Automorphisms and Lie stacks, Commun. Algebra 25 (1997), 2211-2226 (1997). MR 98h:14019

14.
W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Interscience, John Wiley and Sons, New York-London-Sydney, 1966. MR 34:7617

15.
M. Nagata, On the Automorphism Group of $k[x,y]$, Lect. in Math., Kyoto Univ., Kinokuniya, Tokyo, 1972. MR 49:2731

16.
H. Park, A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems, Ph.D. Thesis, Univ. of California, Berkeley, 1995.

17.
V. Shpilrain, J.-T. Yu, Polynomial automorphisms and Gröbner reductions, J. Algebra 197 (1997), 546-558. MR 99c:13055

18.
V. Shpilrain, J.-T. Yu, On generators of polynomial algebras in two commuting or non-commuting variables, J. Pure Appl. Algebra 132 (1998), 309-315. MR 99g:16044

19.
M. K. Smith, Stably tame automorphisms, J. Pure Appl. Algebra 58 (1989), 209-212. MR 90f:13005

20.
D. Wright, The amalgamated free product structure of $GL_{2}(k[X_{1},\ldots ,X_{n}])$ and the weak Jacobian theorem for two variables, J. Pure Appl. Algebra 12 (1978), 235-251. MR 80a:20049

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Additional Information:

Vesselin Drensky
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria
Email: drensky@math.bas.bg

Jie-Tai Yu
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Email: yujt@hkusua.hku.hk

DOI: 10.1090/S0002-9947-00-02617-9
PII: S 0002-9947(00)02617-9
Keywords: Automorphisms of polynomial algebras, tame automorphisms, wild automorphisms, free generators of polynomial algebras
Received by editor(s): March 11, 1999
Posted: October 19, 2000
Additional Notes: The research of the first author was partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.
The research of the second author was partially supported by RGC Grant HKU7126-98P and CRCG Grant 10201869.23067.25500.302.01
Copyright of article: Copyright 2000, American Mathematical Society




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