Endomorphisms of the plane sending linear coordinates to coordinates
Authors:
Charles ChingAn Cheng and Arno van den Essen
Journal:
Proc. Amer. Math. Soc. 128 (2000), 19111915
MSC (1991):
Primary 13B25, 13F20, 14E09, 16W20
Published electronically:
February 25, 2000
MathSciNet review:
1653437
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a field of characteristic zero. We show that an endomorphism of which sends each linear coordinate to a coordinate is an automorphism of .
 [1]
Kossivi
Adjamagbo and Arno
van den Essen, A resultant criterion and formula for the inversion
of a polynomial map in two variables, J. Pure Appl. Algebra
64 (1990), no. 1, 1–6. MR 1055017
(91g:14011), http://dx.doi.org/10.1016/00224049(90)90002Y
 [2]
A.
D. R. Choudary and A.
Dimca, Complex hypersurfaces diffeomorphic to affine spaces,
Kodai Math. J. 17 (1994), no. 2, 171–178. MR 1282208
(95f:14084), http://dx.doi.org/10.2996/kmj/1138039958
 [3]
Arno
van den Essen and Vladimir
Shpilrain, Some combinatorial questions about polynomial
mappings, J. Pure Appl. Algebra 119 (1997),
no. 1, 47–52. MR 1456093
(98g:13007), http://dx.doi.org/10.1016/S00224049(96)000436
 [4]
R. Feßler, A proof of the two dimensional MarkusYamabe stability conjecture, Annales Polonici Mathematici 62 (1995), 4575.
 [5]
A. A. Glutsuk, The complete solution of the Jacobian problem for vector fields on the plane, Communications of the Moscow Mathematical Society, Russian Math. Surveys 49 (1994), 185186, Translated in Uspekhi Mat. Nauk, 3 (1994).
 [6]
Carlos
Gutiérrez, A solution to the bidimensional global asymptotic
stability conjecture, Ann. Inst. H. Poincaré Anal. Non
Linéaire 12 (1995), no. 6, 627–671
(English, with English and French summaries). MR 1360540
(96k:34099b)
 [7]
Z. Jelonek, A solution of the problem of van den Essen and Shpilrain, IMUJ, preprint 1996 /19 Krakow.
 [8]
Shulim
Kaliman, Exotic structures on 𝐂ⁿ and
𝐂*action on 𝐂³, Complex analysis and geometry
(Trento, 1993) Lecture Notes in Pure and Appl. Math., vol. 173,
Dekker, New York, 1996, pp. 299–307. MR 1365979
(97e:14058)
 [9]
James
H. McKay and Stuart
Sui Sheng Wang, An inversion formula for two polynomials in two
variables, J. Pure Appl. Algebra 40 (1986),
no. 3, 245–257. MR 836651
(87j:12003), http://dx.doi.org/10.1016/00224049(86)900447
 [10]
A. Mikhalev, J. Yu and A. Zolotykh, Images of coordinate polynomials, Algebra Colloquium 4 (1997), 159162, preprint 1997.
 [11]
Patrick
J. Rabier, On components of polynomial automorphisms in two
variables, Comm. Algebra 24 (1996), no. 3,
929–937. MR 1374646
(96m:14020), http://dx.doi.org/10.1080/00927879608825612
 [1]
 K. Adjamagbo and A. van den Essen, A resultant criterion and formula for the inversion of a polynomial map in two variables, J. of Pure and Applied Algebra 64 (1990), 16. MR 91g:14011
 [2]
 A. Choudary and A. Dimca, Complex Hypersurfaces diffeomorphic to affine spaces, preprint 1994. MR 95f:14084
 [3]
 A. van den Essen and V. Shpilrain, Some combinatorial questions about polynomial mappings, J. of Pure and Applied Algebra 119 (1997), 4752. MR 98g:13007
 [4]
 R. Feßler, A proof of the two dimensional MarkusYamabe stability conjecture, Annales Polonici Mathematici 62 (1995), 4575.
 [5]
 A. A. Glutsuk, The complete solution of the Jacobian problem for vector fields on the plane, Communications of the Moscow Mathematical Society, Russian Math. Surveys 49 (1994), 185186, Translated in Uspekhi Mat. Nauk, 3 (1994).
 [6]
 C. Gutierrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 627671. MR 96k:34099b
 [7]
 Z. Jelonek, A solution of the problem of van den Essen and Shpilrain, IMUJ, preprint 1996 /19 Krakow.
 [8]
 S. Kaliman, Exotic structures on and actions on , preprint 1994. MR 97e:14058
 [9]
 J. H. McKay and S. S.S. Wang, An inversion formula for two polynomials in two variables, J. of Pure and Applied Algebra 40 (1986), 245257. MR 87j:12003
 [10]
 A. Mikhalev, J. Yu and A. Zolotykh, Images of coordinate polynomials, Algebra Colloquium 4 (1997), 159162, preprint 1997.
 [11]
 P. Rabier, On components of polynomial automorphisms in two variables, Comm. Algebra 24 (1996), 929937. MR 96m:14020
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (1991):
13B25,
13F20,
14E09,
16W20
Retrieve articles in all journals
with MSC (1991):
13B25,
13F20,
14E09,
16W20
Additional Information
Charles ChingAn Cheng
Affiliation:
Department of Mathematics, Oakland University, Rochester, Michigan 48309
Email:
cheng@oakland.edu
Arno van den Essen
Affiliation:
Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands
Email:
essen@sci.kun.nl
DOI:
http://dx.doi.org/10.1090/S0002993900052369
PII:
S 00029939(00)052369
Keywords:
Jacobian conjecture,
automorphism,
endomorphism,
derivation,
algebraically closed field,
coordinate
Received by editor(s):
December 12, 1997
Received by editor(s) in revised form:
September 1, 1998
Published electronically:
February 25, 2000
Communicated by:
Wolmer V. Vasconcelos
Article copyright:
© Copyright 2000 American Mathematical Society
