Selecta Mathematica

, Volume 18, Issue 4, pp 825–837 | Cite as

Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

  • Raf CluckersEmail author
  • Immanuel Halupczok


It is known that a p-adic, locally Lipschitz continuous semi-algebraic function, is piecewise Lipschitz continuous, where finitely many pieces suffice and the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1 (again, with finitely many pieces). We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parameterized versions and in the subanalytic setting.


p-Adic semi-algebraic functions p-Adic subanalytic functions Lipschitz continuous functions p-Adic cell decomposition 

Mathematics Subject Classification (2000)

Primary 03C98 12J25 Secondary 03C60 32Bxx 11S80 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cluckers R.: Presburger sets and p-minimal fields. J. Symb. Logic 68, 153–162 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cluckers R.: Analytic p-adic cell decomposition and integrals. Trans. Am. Math. Soc. 356, 1489–1499 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cluckers, R., Comte, G., Loeser, F.: Lipschitz continuity properties for p-adic semi-algebraic and subanalytic functions. Geom. Funct. Anal. 20(1), 68–87 (2010) arXiv:0904.3853Google Scholar
  4. 4.
    Cluckers, R., Lipshitz, L.: Fields with analytic structure. J. Eur. Math. Soc. 13(4), 1147–1223 (2011) arXiv:0908.2376Google Scholar
  5. 5.
    Cohen P.J.: Decision procedures for real and p-adic fields. Comm. Pure Appl. Math. 22, 131–151 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Denef J.: The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77, 1–23 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Denef J.: p-adic semi-algebraic sets and cell decomposition. J. Reine Angew. Math. 369, 154–166 (1986)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Denef J., van den Dries L.: p-adic and real subanalytic sets. Ann. Math. 128, 79–138 (1988)zbMATHCrossRefGoogle Scholar
  9. 9.
    van den Dries L., Haskell D., Macpherson D.: One-dimensional p-adic subanalytic sets. J. Lond. Math. Soc. 59(2), 1–20 (1999)zbMATHCrossRefGoogle Scholar
  10. 10.
    Glöckner H.: Comparison of some notions of C k-maps in multi-variable non-archimedian analysis. Bull. Belg. Math. Soc. Simon Stevin 14(5), 877–904 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kurdyka, K.: On a Subanalytic Stratification Satisfying a Whitney Property with Exponent 1. Real Algebraic Geometry (Rennes, 1991), pp. 316–322, Lecture Notes in Math., 1524. Springer, Berlin (1992)Google Scholar
  12. 12.
    Macintyre A.: On definable subsets of p-adic fields. J. Symb. Logic 41(3), 605–610 (1976)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Laboratoire Painlevé, CNRS, UMR 8524, Université Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium
  3. 3.Institut für Mathematische Logik und GrundlagenforschungUniversität MünsterMünsterGermany

Personalised recommendations