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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Constants of derivations in polynomial rings over unique factorization domains


Author: M'hammed El Kahoui
Journal: Proc. Amer. Math. Soc. 132 (2004), 2537-2541
MSC (2000): Primary 12H05, 13P10
Published electronically: April 8, 2004
MathSciNet review: 2054777
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Abstract | References | Similar Articles | Additional Information

Abstract: A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any ${\mathcal K}$-derivation of ${\mathcal K}[x,y]$, where ${\mathcal K}$ is a commutative field of characteristic zero, is a polynomial ring in one variable over ${\mathcal K}$. In this paper we give an elementary proof of this theorem and show that it remains true if we replace ${\mathcal K}$ by any unique factorization domain of characteristic zero.


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

  • 1. J. Berson. Derivations on polynomial rings over a domain, Master's thesis, University of Nijmegen, Nijmegen, The Netherlands, 1999.
  • 2. Arno van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. MR 1790619 (2001j:14082)
  • 3. A. Nowicki. Polynomial derivations and their rings of constants. N. Copernicus University Press, Torun, 1994.
  • 4. Andrzej Nowicki and Masayoshi Nagata, Rings of constants for 𝑘-derivations in 𝑘[𝑥₁,\cdots,𝑥_{𝑛}], J. Math. Kyoto Univ. 28 (1988), no. 1, 111–118. MR 929212 (89b:13009)
  • 5. P. van Rossum. Tackling problems on affine space with locally nilpotent derivations on polynomial rings. Ph.D. thesis, University of Nijmegen, The Netherlands, 2001.
  • 6. Abraham Zaks, Dedekind subrings of 𝑘[𝑥₁,\cdots,𝑥_{𝑛}] are rings of polynomials, Israel J. Math. 9 (1971), 285–289. MR 0280471 (43 #6191)
  • 7. O. Zariski, Interprétations algébrico-géométriques du quatorzième problème de Hilbert, Bull. Sci. Math. (2) 78 (1954), 155–168 (French). MR 0065217 (16,398c)

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Additional Information

M'hammed El Kahoui
Affiliation: Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P.O. Box 2390, Marrakech, Morocco
Email: elkahoui@ucam.ac.ma

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07313-7
PII: S 0002-9939(04)07313-7
Keywords: Derivations, ring of constants
Received by editor(s): December 27, 2002
Received by editor(s) in revised form: April 1, 2003
Published electronically: April 8, 2004
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society