A counterexample to a conjecture of Drużkowski and Rusek

• Autorzy: Arno van den Essen
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

polynomial automorphisms; Jacobian Conjecture

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1995
• Tom: 62
• Numer: 2
• Strony: 173-176

Daty

wydano

1995

otrzymano

1995-02-01

Twórcy

autor

Arno van den Essen

• Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands

Bibliografia

• [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.
• [2] L. M. Drużkowski, An effective approach to Keller's Jacobian Conjecture, Math. Ann. 264 (1983), 303-313.
• [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.
• [4] L. M. Drużkowski and K. Rusek, The formal inverse and the Jacobian conjecture, Ann. Polon. Math. 46 (1985), 85-90.
• [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] K. Rusek and T. Winiarski, Polynomial automorphisms of $C^n$, Univ. Iagel. Acta Math. 24 (1984), 143-149.
• [7] A. V. Yagzhev, On Keller's problem, Siberian Math. J. 21 (1980), 747-754.
• [8] J.-T. Yu, On the Jacobian Conjecture: reduction of coefficients, J. Algebra 171 (1995), 515-523.