Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 311–339 | Cite as

Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus

  • Shu KawaguchiEmail author


Let f be a triangular automorphism of the affine N-space of degree d with Jacobian 1 over a \({\mathbb{Q}}\)-algebra R. We introduce a weighted nilpotency index ν(f) for f, and give a bound of deg(f n ) in terms of N, d and ν(f) for all \({n \in \mathbb{Z}}\). When N = 2, our formula, combined with computation of the Hilbert series of certain graded algebras, yields the estimate deg(f n ) ≤ d 2d + 1 for all \({n \in \mathbb{Z}}\). If n varies through all integers, this estimate turns out to be sharp and is related, somewhat unexpectedly, to the Schubert calculus on the Grassmannian G(d − 1, 2d − 2). Numerical computation for small degrees suggests that this estimate restricted to the inverse degree (i.e. n = −1) is also sharp if d ≥ 3.

Mathematics Subject Classification (2000)

08A35 13B25 14J50 14R10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Atiyah M.F., MacDonald I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading, MA (1969)zbMATHGoogle Scholar
  2. 2.
    Bass, H.: The Jacobian conjecture and inverse degrees. Arithmetic and geometry, Vol. II, pp. 65–75, Prog. Math., 36. Birkhäuser, Boston, Mass (1983)Google Scholar
  3. 3.
    Bass H., Connel E.H., Wright D.: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc. (N.S.) 7(2), 287–330 (1982)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cheng C.C.-A., Wang S.S.-S., Yu J.-T.: Degree bounds for inverses of polynomial automorphisms. Proc. Am. Math. Soc. 120(3), 705–707 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Derksen H.: Inverse degrees and the Jacobian Conjecture. Commun. Algebra 22(12), 4793–4794 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Favre C., Jonsson M.: Dynamical compactifications of C 2. Ann. Math. (2) 173(1), 211–248 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fournié M., Furter J.-Ph., Pinchon D.: Computation of the maximal degree of the inverse of a cubic automorphism of the affine plane with Jacobian 1 via Gröbner bases. J. Symbolic Comput. 26(3), 381–386 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fulton, W.: Intersection theory, 2nd edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2. Springer, Berlin (1998)Google Scholar
  9. 9.
    Furter J.-Ph.: On the degree of the inverse of an automorphism of the affine plane. J. Pure Appl. Algebra 130(3), 277–292 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Furter J.-Ph.: On the degree of iterates of automorphisms of the affine plane. Manuscripta Math. 98(2), 183–193 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978)zbMATHGoogle Scholar
  12. 12.
    Grayson D.R., Stillman M.E.: Macaulay2, a software system for research in algebraic geometry, Available at
  13. 13.
    Henrici P.: An algebraic proof of the Lagrange-Bürmann formula. J. Math. Anal. Appl. 8, 218–224 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Jung H.: Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184, 161–174 (1942)MathSciNetGoogle Scholar
  15. 15.
    Kawaguchi S.: Inverse degree of a triangular automorphism of the affine space. Proc. Am. Math. Soc. 141, 3353–3360 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Matsumura, H.: Commutative Ring Theory. Translated from the Japanese by M. Reid, Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, Cambridge (1986)Google Scholar
  17. 17.
    Maubach S.: The automorphism group of \({\mathbb{C}[T]/(T^m)[X_1,\dots,X_n]}\). Commun. Algebra 30(2), 619–629 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    van den Essen, A.:Polynomial Automorphisms (and the Jacobian conjecture), Progress in Mathematics, 190. Birkhäuser, Basel (2000)Google Scholar
  19. 19.
    van den Essen, A.: On Bass’ inverse degree approach to the Jacobian conjecture and exponential automorphisms. Combinatorial and computational algebra (Hong Kong, 1999), pp. 207–214. Contemp. Math., 264. Am. Math. Soc., Providence, RI (2000)Google Scholar
  20. 20.
    van der Kulk W.: On polynomial rings in two variables. Nieuw Arch. Wiskunde (3) 1, 33–41 (1953)zbMATHMathSciNetGoogle Scholar
  21. 21.
    van Rossum, P.: Tackling problems on affine space with locally nilpotent derivations on polynomial rings, Ph.D. thesis, University of Nijmegen (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan

Personalised recommendations