Transformation Groups

, Volume 23, Issue 1, pp 271–297 | Cite as




Let k be a field of positive characteristic p. We introduce the notion of a pseudo-derivation of a k-algebra A, and give a one-to-one correspondence between the set of all pseudo-derivations of A and the set of all p-unipotent automorphisms of A. We classify p-unipotent triangular automorphisms of a polynomial ring k[x, y, z] in three variables over k up to conjugation of automorphisms of k[x, y, z]. We prove that if a p-cyclic group /pℤ acts triangularly on the polynomial ring k[x, y, z], the modular invariant ring k[x, y, z] /pℤ is a hypersurface ring.


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  1. [1]
    H. E. A. Campbell, D. L. Wehlau, Modular Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 139, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VIII, Springer, Berlin, 2011.Google Scholar
  2. [2]
    L. E. Dickson, On Invariants and the Theory of Numbers, The Madison Colloquium (1913, Part 1), Amer. Math. Society, reprinted by Dover, New York, 1966.Google Scholar
  3. [3]
    S. Maubach, Invariants and conjugacy classes of triangular polynomial maps, J. Pure Appl. Algebra 219 (2015), no. 12, 5206–5224.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    M. Miyanishi, Lectures on Curves on Rational and Unirational Surfaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Vol. 60, Narosa, 1978.Google Scholar
  5. [5]
    R. J. Shank, S.A.G.B.I. bases for rings of formal modular seminvariants, Comment. Math. Helv. 73 (1998), no. 4, 548–565.Google Scholar
  6. [6]
    R. J. Shank, Classical covariants and modular invariants, in: Invariant Theory in All Characteristics, CRM Proc. Lecture Notes, Vol. 35, Amer. Math. Society, Providence, RI, 2004, pp. 241–249.Google Scholar
  7. [7]
    Y. Takeda, Artin-Schreier coverings of algebraic surfaces, J. Math. Soc. Japan 41 (1989), no. 3, 415–435.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. L. Wehlau, Invariants for modular representations of a cyclic group of prime order via classical invariant theory, J. Eur. Math. Soc. 15 (2013), no. 3, 775–803.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    W. L. G. Williams, Fundamental systems of formal modular seminvariants of the binary cubic, Trans. Amer. Math. Soc. 22 (1921), no. 1, 56–79.MathSciNetMATHGoogle Scholar
  10. [10]
    D. Wright, Abelian subgroups of Autk(k[X, Y]) and applications to actions on the affine plane, Illinois J. Math. 23 (1979), no. 4, 579–634.MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of EducationShizuoka UniversityShizuokaJapan

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