Tohoku Mathematical Journal

Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism

Shigeru Kuroda

Full-text: Open access


In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the “generalized Shestakov-Umirbaev inequality”, which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we deduce that no tame automorphism of a polynomial ring admits a reduction of type IV.

Article information

Tohoku Math. J. (2), Volume 62, Number 1 (2010), 75-115.

First available in Project Euclid: 31 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
Secondary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]

Polynomial automorphisms tame generators problem


Kuroda, Shigeru. Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism. Tohoku Math. J. (2) 62 (2010), no. 1, 75--115. doi:10.2748/tmj/1270041028.

Export citation


  • 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.
  • A. van den Essen, The solution of the tame generators conjecture according to Shestakov and Umirbaev, Colloq. Math. 100 (2004), 181--194.
  • A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progr. Math. 190, Birkhäuser, Basel, 2000.
  • 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. Wiskunde (3) 1 (1953), 33--41.
  • S. Kuroda, A generalization of the Shestakov-Umirbaev inequality, J. Math. Soc. Japan 60 (2008), 495--510.
  • S. Kuroda, Automorphisms of a polynomial ring which admit reductions of type I, Publ. Res. Inst. Math. Sci. 45 (2009), 907--917.
  • M. Nagata, On automorphism group of $k[x,y]$, Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 5, Kinokuniya Book-Store Co., Ltd., Tokyo, 1972.
  • 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.