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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Artin, M.: Algebraic approximation of structures over complete local rings. Publ. IHES 36, 23–58 (1969)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bilski, M.: Approximation of analytic sets by Nash tangents of higher order. Math. Z. 256(4), 705–716 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bochnak, J., Kucharz, W.: Local algebraicity of analytic sets. J. Reine Angew. Math. 352, 1–14 (1984)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferrarotti, M., Fortuna, E., Wilson, L.: Local algebraic approximation of semianalytic sets. Proc. Am. Math. Soc. 143, 13–23 (2015)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kucharz, W.: Power series and smooth functions equivalent to a polynomial. Proc. Am. Math. Soc. 98(3), 527–533 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kurdyka, K.: Ensembles semi-algébriques symétriques par arcs. Math. Ann. 282, 445–462 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lempert, L.: Algebraic approximations in analytic geometry. Invent. Math. 121, 335–354 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mostowski, T.: Topological equivalence between analytic and algebraic sets. Bull. Pol. Acad. Sci. Math. 32(7–8), 393–400 (1984)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Parusiński, A., Paunescu, L.: Arcwise analytic stratification, Whitney fibering conjecture and Zariski equisingularity. Adv. Math. 309, 254–305 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Popescu, D.: General Néron desingularization. Nagoya Math. J. 100, 97–126 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Quarez, R.: The Artin conjecture for \({\mathbb{Q}}\)-algebras. Rev. Mat. Univ. Complut. Madr 10(2), 229–263 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Samuel, P.: Algébricité de certains points singuliers algébroïdes. J. Math. Pures Appl. 35, 1–6 (1956)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Schappacher, N.: L’inégalité de Łojasiewicz ultramétrique. CR Acad. Sci. Paris Sér. I Math. 296(10), 439–442 (1983)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tworzewski, P.: Intersections of analytic sets with linear subspaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17, 227–271 (1990)MathSciNetzbMATHGoogle 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)CrossRefzbMATHGoogle 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)zbMATHGoogle 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

Personalised recommendations