Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Coordinates in two variables over a $\mathbb{Q} $-algebra


Authors: Arno van den Essen and Peter van Rossum
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 1691-1703
MSC (2000): Primary 13B25, 14J70, 14R10
DOI: https://doi.org/10.1090/S0002-9947-04-03492-0
Published electronically: January 6, 2004
MathSciNet review: 2031037
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies coordinates in two variables over a $\mathbb{Q} $-algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary $\mathbb{Q} $-algebras.


References [Enhancements On Off] (What's this?)

  • [AE90] K. Adjamagbo and A. van den Essen, A resultant criterion and formula for the inversion of a polynomial map in two variables, J. Pure Appl. Alg. 64 (1990), 1-6. MR 91g:14011
  • [AHE72] Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Alg. 23 (1972), 310-342. MR 46:5300
  • [AM75] Shreeram S. Abhyankar and Tzuong-tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166. MR 52:407
  • [BD93] S. M. Bhatwadekar and Amartya K. Dutta, On residual variables and stably polynomial algebras, Comm. Alg. 21 (1993), no. 2, 635-645. MR 93k:13028
  • [BD94] -, Linear planes over a discrete valuation ring, J. Alg. 166 (1994), 393-405. MR 95e:13006
  • [BEM01] Joost Berson, Arno van den Essen, and Stefan Maubach, Derivations having divergence zero on $R[X,Y]$, Isr. J. Math. 124 (2001), 115-124. MR 2002f:13057
  • [Ber99] Joost Berson, Derivations of polynomial rings over a domain, Master's thesis, University of Nijmegen, June 1999.
  • [Ber02] -, Stably tame coordinates, J. Pure Apl. Alg. 170 (2002), 131-243. MR 2003e:13009
  • [CE00] Charles Ching-An Cheng and Arno van den Essen, Endomorphisms of the plane sending linear coordinates to coordinates, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1911-1915. MR 2000m:14072
  • [CK92] J. Chadzynski and T. Krasinski, On the Lojasiewicz exponent at infinity for polynomial mappings of $\mathbb{C} ^2$ into $\mathbb{C} ^2$ and components of polynomial automorphisms of $\mathbb{C} ^2$, Ann. Polon. Math. 57 (1992), no. 3, 291-302. MR 94g:14004
  • [CMW95] C. C. Cheng, J. H. McKay, and S. S.-S. Wang, Younger mates and the Jacobian Conjecture, Proc. Amer. Math. Soc. 123 (1995), 2939-2947. MR 95m:14014
  • [DY01] D. Drensky and 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
  • [ER01] Arno van den Essen and Peter van Rossum, A class of counterexamples to the cancellation problem for arbitrary rings, Ann. Polon. Math. 76 (2001), 89-93. MR 2002d:13025
  • [Ess93] Arno van den Essen, Locally nilpotent derivations and their applications III, J. Pure Apl. Alg. 98 (1995), 15-23. MR 96a:13006
  • [Ess00] -, Polynomial automorphisms and the Jacobian Conjecture, Progress in Mathematics, vol. 190, Birkhäuser-Verlag, Basel-Boston-Berlin, 2000. MR 2001j:14082
  • [EV01] E. Edo and S. Vénéreau, Length 2 variables and transfer, Annales Polonici Math. 76 (2001), 67-76. MR 2002f:14080
  • [MYZ97] A. Mikhalev, J.-T. Yu, and A. Zolotykh, Images of coordinate polynomials, Alg. Coll. 4 (1997), no. 2, 159-162. MR 2000a:13036
  • [Nag72] M. Nagata, On the automorphism group of $k[X,Y]$, Kyoto Univ. Lectures in Math. 5 (1972). MR 49:2731
  • [Rus76] P. Russell, Simple birational extensions of two dimensional affine rational domains, Comp. Math. 33 (1976), 197-208. MR 55:2943
  • [Sat76] Avinash Sathaye, On linear planes, Proc. Amer. Math. Soc. 56 (1976), 1-7. MR 53:13227
  • [Sha92] Anant R. Shastri, Polynomial representations of knots, Tôhoku Math. J. 44 (1992), 11-17. MR 92k:57016
  • [Suz74] M. Suzuki, Propriétés topologiques des polynômes de deux variables complex, et automoprhisms algébriques de l'espace $\mathbb{C} ^2$, J. Math. Soc. Japan 26 (1974), no. 3, 241-257. MR 49:3188
  • [Wri81] David Wright, On the Jacobian Conjecture, Illinois J. Math. 25 (1981), no. 3, 423-440. MR 83a:12032

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13B25, 14J70, 14R10

Retrieve articles in all journals with MSC (2000): 13B25, 14J70, 14R10


Additional Information

Arno van den Essen
Affiliation: Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands
Email: essen@math.kun.nl

Peter van Rossum
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: petervr@nmsu.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03492-0
Keywords: Coordinates, locally nilpotent derivations, embeddings
Received by editor(s): February 8, 2001
Published electronically: January 6, 2004
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society