Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven


Author: Gene Freudenburg
Journal: Proc. Amer. Math. Soc. 135 (2007), 51-57
MSC (2000): Primary 13A50, 14R20
DOI: https://doi.org/10.1090/S0002-9939-06-08532-7
Published electronically: July 28, 2006
MathSciNet review: 2280174
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A family of $ \mathbb{G}_a$-actions on affine space $ \mathbb{A}^m$ is constructed, each having a non-finitely generated ring of invariants ($ m\ge 6$). Because these actions are of small degree, they induce linear actions of unipotent groups $ \mathbb{G}_a^n\rtimes\mathbb{G}_a$ on $ \mathbb{A}^{2n+3}$ for $ n\ge 4$, and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group $ \mathbb{G}_a^4\rtimes\mathbb{G}_a$ acting linearly on $ \mathbb{A}^{11}$.


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

  • 1. A. A'Campo-Neuen, Note on a counterexample to Hilbert's fourteenth problem given by P. Roberts, Indag. Math., N.S. 5 (1994), 253-257. MR 1298772 (95k:13003)
  • 2. D. Daigle and G. Freudenburg, A counterexample to Hilbert's Fourteenth Problem in dimension five, J. Algebra 221 (1999), 528-535. MR 1728394 (2000i:13029)
  • 3. H. Derksen and G. Kemper, Computational Invariant Theory, Springer Verlag, Berlin, Heidelberg, New York, 2002. MR 1918599 (2003g:13004)
  • 4. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhauser, Boston, 2000. MR 1790619 (2001j:14082)
  • 5. G. Freudenburg, Locally Nilpotent Derivations and $ {\mathbb{G}}_a$-Actions, Springer-Verlag, New York, (to appear).
  • 6. J. Khoury, On some properties of elementary derivations in dimension six, J. Pure Appl. Algebra 156 (2001), 69-79. MR 1807016 (2001m:13045)
  • 7. H. Kojima and M. Miyanishi, On P. Roberts' counterexample to the fourteenth problem of Hilbert, J. Pure Appl. Algebra 122 (1997), 247-268. MR 1481092 (98j:13008)
  • 8. K. Kurano, Positive characteristic finite generatiion of symbolic Rees algebra and Roberts' counterexamples to the fourteenth problem of Hilbert, Tokyo J. Math. 16 (1993), 473-496. MR 1247667 (94k:13004)
  • 9. S. Kuroda, A counterexample to the Fourteenth Problem of Hilbert in dimension four, J. Algebra 279 (2004), 126-134. MR 2078390 (2005f:13022)
  • 10. -, A generalization of Roberts' counterexample to the fourteenth problem of Hilbert, Tohoku Math. J. 56 (2004), 501-522. MR 2097158 (2005i:13008)
  • 11. -, A counterexample to the Fourteenth Problem of Hilbert in dimension three, Michigan Math. J. 53 (2005), 123-132. MR 2125538 (2005k:13043)
  • 12. S. Maubach, Triangular monomial derivations on $ k[x_1,x_2,x_3,x_4]$ have kernel generated by at most four elements, J. Pure Appl. Algebra 153 (2000), 165-170. MR 1780741 (2001g:13057)
  • 13. S. Mukai, Geometric realization of T-shaped root systems and counterexamples to Hilbert's fourteenth problem, Algebraic Transformation Groups and Algebraic Varieties, Springer-Verlag, Berlin, 2004, Encyclopaedia Math. Sci. 132, 123-129. MR 2090672 (2005h:13008)
  • 14. M. Nagata, On the Fourteenth Problem of Hilbert, Proc. I.C.M. 1958, Cambridge University Press, 1960, pp. 459-462. MR 0116056 (22:6851)
  • 15. P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem, J. Algebra 132 (1990), 461-473. MR 1061491 (91j:13006)
  • 16. R. Steinberg, Nagata's example, Algebraic Groups and Lie Groups, Cambridge University Press, 1997, pp. 375-384.
  • 17. R. Tanimoto, Linear counterexamples to the fourteenth problem of Hilbert, J. Algebra 275 (2004), 331-338. MR 2047451 (2005g:13012)
  • 18. O. Zariski, Interpretations algebrico-geometriques du quatorzieme problem de Hilbert, Bull. Sci. Math. 78 (1954), 155-168. MR 0065217 (16:398c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A50, 14R20

Retrieve articles in all journals with MSC (2000): 13A50, 14R20


Additional Information

Gene Freudenburg
Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008
Email: gene.freudenburg@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-06-08532-7
Keywords: Hilbert's Fourteenth Problem, invariant theory, locally nilpotent derivations
Received by editor(s): August 10, 2005
Published electronically: July 28, 2006
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society