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Quasi-translations and counterexamples to the Homogeneous Dependence Problem

Author: Michiel de Bondt
Journal: Proc. Amer. Math. Soc. 134 (2006), 2849-2856
MSC (2000): Primary 14R15, 14R20, 14R99
Published electronically: May 2, 2006
MathSciNet review: 2231607
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Abstract: In this article, the author gives counterexamples to the Linear Dependence Problem for Homogeneous Nilpotent Jacobians for dimension 5 and up. This problem has been formulated as a conjecture/problem by several authors in connection to the Jacobian conjecture. In dimension 10 and up, cubic counterexamples are given.

In the construction of these counterexamples, the author makes use of so-called quasi-translations, a special type of invertible polynomial maps. Quasi-translations can also be seen as a special type of locally nilpotent derivations.

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

  • 1. H. Bass, E. Connel and D. Wright, The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse, Bull. of the AMS, 7 (1982), 287-330. MR 0663785 (83k:14028)
  • 2. M. de Bondt and A. van den Essen, A reduction of the Jacobian Conjecture to the symmetric case, Proc. Amer. Math. Soc., 133 (2005), 2201-2205. MR 2138860
  • 3. M. de Bondt and A. van den Essen, Singular Hessians, J. Algebra, 282 (2004), 195-204.MR 2095579 (2005i:13031)
  • 4. M. de Bondt and A. van den Essen, The Jacobian Conjecture: linear triangularization for homogeneous polynomial maps in dimension three, J. Algebra, 294 (2005), 294-306. MR 2179727
  • 5. M. de Bondt, Homogeneous quasi-translations and an article of P. Gordan and M. Nöther, Report 0417, Dep. Math., Radboud University Nijmegen.
  • 6. A. Cima, A. Gasull and F. Mañosas, The discrete Markus-Yamabe problem, Nonlinear Analysis: Theory, Methods & Applications, 35 (1999), no. 3, pp. 343-354.MR 1643454 (2000j:37030)
  • 7. A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Vol. 190 in Progress in Mathematics, Birkhäuser, 2000. MR 1790619 (2001j:14082)
  • 8. G. Freudenburg, Actions of $ G_a$ on $ {\mathbb{A}}^3$ defined by homogeneous derivations, J. of Pure and Applied Algebra, 126 (1998), 169-181. MR 1600530 (98k:14068)
  • 9. P. Gordan and M. Nöther, Über die algebraische Formen, deren Hesse'sche Determinante identisch verschwindet, Mathematische Annalen, 10 (1876), 547-568. MR 1509898
  • 10. E. Hubbers, The Jacobian Conjecture: Cubic homogeneous maps in Dimension Four, Master's thesis, Univ. of Nijmegen, 1994.
  • 11. G.H. Meisters, Polyomorphisms conjugate to Dilations, In Automorphisms of Affine Spaces, Curaçao, July 4-8, 1994, Caribean Mathematics Foundation, Kluwer Academic Press, 1995, Proceedings of the conference `Invertible Polynomial maps'. MR 1352691 (97m:14020)
  • 12. C. Olech, On Markus-Yamabe stability conjecture, Proc. of the Intern. Meeting on Ordinary Differential Equations and their Applications, Univ. of Florence, 1995, pp. 127-137.
  • 13. K. Rusek, Polynomial Automorphisms, preprint 456, Institute of Mathematics, Polish Academy of Sciences, IMPAN, Sniadeckich 8, P.O. Box 137, 00-950 Warsaw, Poland, May 1989.
  • 14. Z. Wang, Homogeneization of Locally Nilpotent Derivations and an Application to $ k[X,Y,Z]$, J. Pure Appl. Algebra, 196 (2005), 323-337. MR 2110528 (2005j:13025)

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Additional Information

Michiel de Bondt
Affiliation: Department of Mathematics, Radboud University, Postbus 9010, 6500 GL Nijmegen, The Netherlands

Keywords: Jacobian (conjecture), quasi-translation, linear dependence problem
Received by editor(s): October 5, 2004
Received by editor(s) in revised form: April 27, 2005
Published electronically: May 2, 2006
Additional Notes: The author was supported by the Netherlands Organization of Scientific Research (NWO)
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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