Mathematische Annalen

, 343:901 | Cite as

Plane polynomial automorphisms of fixed multidegree

Article

Abstract

Let \({\mathcal{G}}\) be the group of polynomial automorphisms of the complex affine plane. On one hand, \({\mathcal {G}}\) can be endowed with the structure of an infinite dimensional algebraic group (see Shafarevich in Math USSR Izv 18:214–226, 1982) and on the other hand there is a partition of \({\mathcal{G}}\) according to the multidegree (see Friedland and Milnor in Ergod Th Dyn Syst 9:67–99, 1989). Let \({{\mathcal{G}}_d}\) denote the set of automorphisms whose multidegree is equal to d. We prove that \({{\mathcal{G}}_d}\) is a smooth, locally closed subset of \({\mathcal{G}}\) and show some related results. We give some applications to the study of the varieties \({{\mathcal G}_{= \, m}}\) (resp. \({{\mathcal G}_{\leq \, m}}\)) of automorphisms whose degree is equal to m (resp. is less than or equal to m).

References

  1. 1.
    Abhyankar S.S., Moh T.T.: Embeddings of the line in the plane. J. Reine Angew. Math. 276, 148–166 (1975)MATHMathSciNetGoogle Scholar
  2. 2.
    Bass H., Connell E., Wright D.: The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. A.M.S. 7, 287–330 (1982)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Edo E., Furter J.-P.: Some families of polynomial automorphisms. J. Pure Appl. Algebra 194, 263–271 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    van den Essen, A.: Polynomial automorphisms and the Jacobian conjecture, Progress in Math. (Boston Mass.), vol. 190. Birkhäuser, Basel (2000)Google Scholar
  5. 5.
    Freudenburg, G.: Algebraic theory of locally nilpotent derivations. Encyclopaedia of Mathematical Sciences, vol. 136. Invariant Theory and Algebraic Transformation Groups, VII. Springer, Berlin (2006)Google Scholar
  6. 6.
    Friedland S., Milnor J.: Dynamical properties of plane polynomial automorphisms. Ergod. Th Dyn. Syst. 9, 67–99 (1989)MATHMathSciNetGoogle Scholar
  7. 7.
    Furter J.-P.: On the variety of automorphisms of the affine plane. J. Algebra 195, 604–623 (1997)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Furter J.-P.: On the length of polynomial automorphisms of the affine plane. Math. Ann. 322, 401–411 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Furter, J.-P.: Polynomial composition rigidity and plane polynomial automorphisms, submitted for publication. Available at http://perso.univ-lr.fr/jpfurter/
  10. 10.
    Grothendieck, A., Dieudonné, J.: Eléments de géométrie algébrique, II Etude globale élémentaire de quelques classes de morphismes. Publ. Math. IHES 8 (1961)Google Scholar
  11. 11.
    Humphreys, J.E.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Math., vol. 21. Springer, Heidelberg (1981)Google Scholar
  12. 12.
    Jung H.W.E.: Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184, 161–174 (1942)MathSciNetGoogle Scholar
  13. 13.
    Kambayashi T.: Pro-affine algebras, ind-affine groups and the Jacobian problem. J. Algebra 185(2), 481–501 (1996)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kambayashi T.: Some basic results on pro-affine algebras and ind-affine schemes. Osaka J. Math. 40(3), 621–638 (2003)MATHMathSciNetGoogle Scholar
  15. 15.
    Kraft, H.: Geometrische Methoden in der Invariantentheorie. (German) [Geometrical methods in invariant theory] Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig (1984)Google Scholar
  16. 16.
    van der Kulk W.: On polynomial rings in two variables. Nieuw. Arch. Wisk. 1(3), 33–41 (1953)MATHGoogle Scholar
  17. 17.
    Makar-Limanov, L.: Locally nilpotent derivations, a new ring invariant and applications. Lecture notes, (1998). Available at http://www.math.wayne.edu/~lml (1998)
  18. 18.
    Mumford, D.: The red book of varieties and schemes (2nd, expanded edn.). Lecture Notes in Math., vol. 1358. Springer, Berlin (1999)Google Scholar
  19. 19.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer, Berlin (1994)Google Scholar
  20. 20.
    Nagata M.: Imbedding of an abstract variety in a complete variety. J. Math. Kyoto Univ. 2, 1–10 (1962)MATHMathSciNetGoogle Scholar
  21. 21.
    Nagata, M.: On automorphism group of k[x, y]. Lecture Notes in Math., vol. 5. Kyoto Univ. (1972)Google Scholar
  22. 22.
    Rentschler R.: Opérations du groupe additif sur le plan affine (French). C. R. Acad. Sci. Paris Sér. A–B 267, A384–A387 (1968)MathSciNetGoogle Scholar
  23. 23.
    Russell P., Sathaye A.: On finding and cancelling variables in k[X, Y, Z]. J. Algebra 57(1), 151–166 (1979)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sathaye A.: On linear planes. Proc. Am. Math. Soc. 56, 1–7 (1976)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Shafarevich, I.R.: On some infinite-dimensional groups. Rend. Mat. e Appl. (5) 25, (1–2) 208–212 (1966)Google Scholar
  26. 26.
    Shafarevich I.R.: On some infinite-dimensional groups II. Math. USSR Izv. 18, 214–226 (1982)Google Scholar
  27. 27.
    Shafarevich I.R.: Letter to the editors: “On some infinite-dimensional groups II” (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 59(3), 224 (1995)MathSciNetGoogle Scholar
  28. 28.
    Suzuki M.: Propriétés topologiques des polynômes de deux variables complexes et automorphismes algébriques de l’espace C2. J. Math. Soc. Japan. 26, 241–257 (1974)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Wright D.: Abelian subgroups of Autk(k[X,Y]) and applications to actions on the affine plane. Illinois J. Math. 23.4, 579–634 (1979)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of La RochelleLa RochelleFrance

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