Journal of the Mathematical Society of Japan

A generalization of the Shestakov-Umirbaev inequality

Shigeru KURODA

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We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the Tame Generators Problem on the automorphism group of a polynomial ring. As an application, we give a new necessary condition for endomorphisms of a polynomial ring to be invertible, which implies Jung's theorem in case of two variables.

Article information

J. Math. Soc. Japan, Volume 60, Number 2 (2008), 495-510.

First available in Project Euclid: 30 May 2008

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Zentralblatt MATH identifier

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Secondary: 12H05: Differential algebra [See also 13Nxx]

polynomial automorphism Tame Generators Problem


KURODA, Shigeru. A generalization of the Shestakov-Umirbaev inequality. J. Math. Soc. Japan 60 (2008), no. 2, 495--510. doi:10.2969/jmsj/06020495.

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  • A. van den Essen, L. Makar-Limanov and R. Willems, Remarks on Shestakov-Umirbaev, Report 0414, Radboud University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands, 2004.
  • H. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math., 184 (1942), 161–174.
  • W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3), 1 (1953), 33–41.
  • M. Nagata, On Automorphism Group of $k[x,y]$, Lectures in Mathematics, Department of Mathematics, Kyoto University, 5, Kinokuniya Book-Store Co. Ltd., Tokyo, 1972.
  • M. K. Smith, Stably tame automorphisms, J. Pure Appl. Algebra, 58 (1989), 209–212.
  • I. Shestakov and U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc., 17 (2004), 181–196.
  • I. Shestakov and U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., 17 (2004), 197–227.
  • S. Vénéreau, A parachute for the degree of a polynomial in algebraically independent ones.1.