Advertisement

Israel Journal of Mathematics

, Volume 190, Issue 1, pp 349–363 | Cite as

Schanuel property for additive power series

  • Piotr Kowalski
Article

Abstract

We prove a version of Schanuel’s Conjecture for a field of Laurent power series in positive characteristic replacing ℂ and a non-algebraic additive power series replacing the exponential map.

Keywords

Power Series Algebraic Group Smooth Point Algebraic Degree Algebraic Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.-P. Allouche, M. Mendès-France and A. J. van der Poorten, Indépendance algébrique de certaines séries formelles Bulletin de la Société Mathématique se France 116 (1988), 449–454.zbMATHGoogle Scholar
  2. [2]
    J. Ax, On Schanuel’s conjectures, Annals of Mathematics 93 (1971), 252–268.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, American Journal of Mathematics 94 (1972), 1195–1204.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Bays, J. Kirby and A. Wilkie, A Schanuel property for exponentially transcendental powers, The Bulletin of the London Mathematical Society 42 (2010), 917–922.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D. Bertrand, Schanuel’s conjecture for non-isoconstant elliptic curves over function fields, in Model theory with applications to algebra and analysis. Vol. 1, London Math. Soc. Lecture Note Ser. vol. 349, Cambridge University Press, Cambridge, 2008, pp. 41–62.Google Scholar
  6. [6]
    W. D. Brownawell, Transcendence in positive characteristic, in Number theory (Tiruchirapalli, 1996), Vol. 210 of Contemporary Mathematics, American Mathematical Society, Providence, RI, 1998, pp. 317–332.CrossRefGoogle Scholar
  7. [7]
    W. D. Brownawell and K. Kubota, Algebraic independence of Weierstrass functions Acta Arithmetica 33 (1977), 111–149.zbMATHMathSciNetGoogle Scholar
  8. [8]
    C. Chevalley, Théorie des groupes de Lie, vol II, Groupes algébriques, Springer, Berlin, 1951.Google Scholar
  9. [9]
    L. Denis, Indépendance algébrique et exponentielle de Carlitz, Acta Arithmetica 69 (1995), 75–89.zbMATHMathSciNetGoogle Scholar
  10. [10]
    V. G. Drinfeld, Elliptic modules (Russian), MatematicheskiĭSbornik. Novaya Seriya 94 (1974), 594–627. English translation: Mathematics of the USSR-Sbornik 23 (1976), 561–592.MathSciNetGoogle Scholar
  11. [11]
    D. Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry, Springer, Berlin, 1996.Google Scholar
  12. [12]
    J. Kirby, The theory of the exponential differential equations of semiabelian varieties, Selecta Mathematica 15 (2009), 445–486.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    P. Kowalski, A note on a theorem of Ax, Annals of Pure and Applied Logic 156 (2008), 96–109.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.Google Scholar
  15. [15]
    M. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Inventiones Mathematicae 171 (2008), 123–174.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    P. Vojta, Jets via Hasse-Schmidt derivations in Diophantine Geometry, Proceedings (U. Zannier, ed.), Edizioni della Normale, Pisa, 2006, pp. 335–361.Google Scholar
  17. [17]
    J. Yu, Transcendence and Drinfeld modules, Inventiones Mathematicae 83 (1986), 507–517.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

Personalised recommendations