A counterexample to a conjecture of Drużkowski and Rusek

Arno van den Essen

Annales Polonici Mathematici (1995)

  • Volume: 62, Issue: 2, page 173-176
  • ISSN: 0066-2216

Abstract

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Let F = X + H be a cubic homogeneous polynomial automorphism from ℂ n to ℂ n . Let p be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that d e g F - 1 ≤ 3 p - 1 . We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.

How to cite

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Arno van den Essen. "A counterexample to a conjecture of Drużkowski and Rusek." Annales Polonici Mathematici 62.2 (1995): 173-176. <http://eudml.org/doc/262620>.

@article{ArnovandenEssen1995,
abstract = {Let F = X + H be a cubic homogeneous polynomial automorphism from $ℂ^n$ to $ℂ^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg F^\{-1\} ≤ 3^\{p-1\}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.},
author = {Arno van den Essen},
journal = {Annales Polonici Mathematici},
keywords = {polynomial automorphisms; Jacobian Conjecture; polynomial automorphism; Jacobian matrix},
language = {eng},
number = {2},
pages = {173-176},
title = {A counterexample to a conjecture of Drużkowski and Rusek},
url = {http://eudml.org/doc/262620},
volume = {62},
year = {1995},
}

TY - JOUR
AU - Arno van den Essen
TI - A counterexample to a conjecture of Drużkowski and Rusek
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 2
SP - 173
EP - 176
AB - Let F = X + H be a cubic homogeneous polynomial automorphism from $ℂ^n$ to $ℂ^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg F^{-1} ≤ 3^{p-1}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.
LA - eng
KW - polynomial automorphisms; Jacobian Conjecture; polynomial automorphism; Jacobian matrix
UR - http://eudml.org/doc/262620
ER -

References

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  1. [1] H. Bass, E. Connell, and D. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330. Zbl0539.13012
  2. [2] L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983), 303-313. Zbl0504.13006
  3. [3] L. M. Drużkowski, The Jacobian Conjecture: some steps towards solution, in: Automorphisms of Affine Spaces, Proc. Conf. 'Invertible Polynomial Maps', Curaçao, July 4-8, 1994, A. R. P. van den Essen (ed.), Caribbean Mathematics Foundation, Kluwer Academic Publishers, 1995, 41-54. Zbl0839.13012
  4. [4] L. M. Drużkowski and K. Rusek, The formal inverse and the Jacobian conjecture, Ann. Polon. Math. 46 (1985), 85-90. Zbl0644.12010
  5. [5] E.-M. G. M. Hubbers, The Jacobian Conjecture: cubic homogeneous maps in dimension four, master thesis, Univ. of Nijmegen, February 17, 1994; directed by A. R. P. van den Essen. 
  6. [6] K. Rusek and T. Winiarski, Polynomial automorphisms of C n , Univ. Iagel. Acta Math. 24 (1984), 143-149. 
  7. [7] A. V. Yagzhev, On Keller's problem, Siberian Math. J. 21 (1980), 747-754. Zbl0466.13009
  8. [8] J.-T. Yu, On the Jacobian Conjecture: reduction of coefficients, J. Algebra 171 (1995), 515-523. Zbl0816.13017

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