Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1361–1375 | Cite as

Higher order approximation of analytic sets by topologically equivalent algebraic sets

  • Marcin Bilski
  • Krzysztof Kurdyka
  • Adam Parusiński
  • Guillaume Rond


It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.


Topological equivalence of singularities Artin approximation Zariski equisingularity 

Mathematics Subject Classification

32S05 32S15 13B40 


  1. 1.
    Akbulut, S., King, H.: On approximating submanifolds by algebraic sets and a solution to the Nash conjecture. Invent. Math. 107, 87–98 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artin, M.: Algebraic approximation of structures over complete local rings. Publ. IHES 36, 23–58 (1969)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Artin, M., Mazur, B.: On periodic points. In: Annals of Mathematics, Second series, vol. 81, no. 1, pp. 82–99 (1965)Google Scholar
  4. 4.
    Bilski, M., Parusiński, A., Rond, G.: Local topological algebraicity of analytic function germs. J. Algebraic Geom. 26, 177–197 (2017)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bilski, M.: Approximation of analytic sets by Nash tangents of higher order. Math. Z. 256(4), 705–716 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bilski, M.: Higher order approximation of complex analytic sets by algebraic sets. Bull. Sci. Math. 139(2), 198–213 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bochnak, J.: Algebraicity versus analyticity, Rocky Mountain. J. Math. 14(4), 863–880 (1984)MathSciNetMATHGoogle Scholar
  8. 8.
    Bochnak, J., Kucharz, W.: Local algebraicity of analytic sets. J. Reine Angew. Math. 352, 1–14 (1984)MathSciNetMATHGoogle Scholar
  9. 9.
    Bochnak, J., Coste, M., Roy, M.-F.: Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin (1998)Google Scholar
  10. 10.
    Braun, R.W., Meise, R., Taylor, B.A.: Higher order tangents to analytic varieties along curves. Can. J. Math. 55, 64–90 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ferrarotti, M., Fortuna, E., Wilson, L.: Local algebraic approximation of semianalytic sets. Proc. Am. Math. Soc. 143, 13–23 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ferrarotti, M., Fortuna, E., Wilson, L.: Algebraic approximation preserving dimension. In: Annals of Mathematics Pura Applcations, Fourth series, vol. 196, no. 2, pp. 519–531 (2017)Google Scholar
  13. 13.
    Greenberg, M.J.: Rational points in Henselian discrete valuation rings. Publ. Math. IHES 31, 59–64 (1966)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kucharz, W.: Power series and smooth functions equivalent to a polynomial. Proc. Am. Math. Soc. 98(3), 527–533 (1986)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kurdyka, K.: Ensembles semi-algébriques symétriques par arcs. Math. Ann. 282, 445–462 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lempert, L.: Algebraic approximations in analytic geometry. Invent. Math. 121, 335–354 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mostowski, T.: Topological equivalence between analytic and algebraic sets. Bull. Pol. Acad. Sci. Math. 32(7–8), 393–400 (1984)MathSciNetMATHGoogle Scholar
  18. 18.
    Parusiński, A., Paunescu, L.: Arcwise analytic stratification, Whitney fibering conjecture and Zariski equisingularity. Adv. Math. 309, 254–305 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Płoski, A.: Note on a theorem of M. Artin. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys 22, 1107–1109 (1974)MathSciNetMATHGoogle Scholar
  20. 20.
    Popescu, D.: General Néron desingularization. Nagoya Math. J. 100, 97–126 (1985)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Quarez, R.: The Artin conjecture for \({\mathbb{Q}}\)-algebras. Rev. Mat. Univ. Complut. Madr 10(2), 229–263 (1997)MathSciNetMATHGoogle Scholar
  22. 22.
    Samuel, P.: Algébricité de certains points singuliers algébroïdes. J. Math. Pures Appl. 35, 1–6 (1956)MathSciNetMATHGoogle Scholar
  23. 23.
    Schappacher, N.: L’inégalité de Łojasiewicz ultramétrique. CR Acad. Sci. Paris Sér. I Math. 296(10), 439–442 (1983)MathSciNetMATHGoogle Scholar
  24. 24.
    Spivakovsky, M.: A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms. J. Am. Math. Soc. 12(2), 381–444 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Swan, R.: Néron-Popescu desingularization, Algebra and geometry (Taipei, 1995), pp. 135-192, Lect. Algebra Geom., 2, Internat. Press, Cambridge, (1998)Google Scholar
  26. 26.
    Tougeron, J.-C.: Solutions d’un système d’équations analytiques réelles et applications. Ann. Inst. Fourier 26, 109–135 (1976)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tworzewski, P.: Intersections of analytic sets with linear subspaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17, 227–271 (1990)MathSciNetMATHGoogle Scholar
  28. 28.
    Varchenko, A.N.: Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings. Math. USSR Izviestija 6, 949–1008 (1972)CrossRefGoogle Scholar
  29. 29.
    Varchenko, A.N.: The relation between topological and algebro-geometric equisingularities according to Zariski. Funkcional. Anal. Appl. 7, 87–90 (1973)CrossRefMATHGoogle Scholar
  30. 30.
    Varchenko, A.N.: Algebro-geometrical equisingularity and local topological classification of smooth mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), 1, pp 427–431. Canad. Math. Congress, Montreal, Que., (1975)Google Scholar
  31. 31.
    Whitney, H.: Local properties of analytic varieties, in 1965 Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 205–244. Princeton Univ. Press, Princeton (1965)Google Scholar
  32. 32.
    Whitney, H.: Complex analytic varieties. Addison-Wesley Publ. Co., Reading, Massachusetts (1972)MATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  • Marcin Bilski
    • 1
  • Krzysztof Kurdyka
    • 2
  • Adam Parusiński
    • 3
  • Guillaume Rond
    • 4
  1. 1.Department of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.Université Savoie Mont Blanc, CNRS, LAMA, UMR 5127Le Bourget-du-LacFrance
  3. 3.Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351NiceFrance
  4. 4.Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373MarseilleFrance

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