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Izv. Akad. Nauk SSSR Ser. Mat., 1972, Volume 36, Issue 5, Pages 957–1019 (Mi izv2343)  

This article is cited in 21 scientific papers (total in 21 papers)

Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings

A. N. Varchenko

Abstract: In this paper we consider families of complex or real algebraic varieties. We prove that for almost all values of the parameters both the topology of the variety and its position in space will be the same. The set of singular values of the parameters is calculated constructively. In this paper we also isolate a class of families of polynomial mappings. For such families we prove the topological equivalence of almost all the mappings included in them. These results are applied to a proof of Zariski's theorem on the fundamental group of the complement to an algebraic hypersurface.

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English version:
Mathematics of the USSR-Izvestiya, 1972, 6:5, 949–1008

Bibliographic databases:

UDC: 513.6
MSC: Primary 14A10, 14E15; Secondary 14F05
Received: 15.02.1972

Citation: A. N. Varchenko, “Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings”, Izv. Akad. Nauk SSSR Ser. Mat., 36:5 (1972), 957–1019; Math. USSR-Izv., 6:5 (1972), 949–1008

Citation in format AMSBIB
\by A.~N.~Varchenko
\paper Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1972
\vol 36
\issue 5
\pages 957--1019
\jour Math. USSR-Izv.
\yr 1972
\vol 6
\issue 5
\pages 949--1008

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. I. Arnol'd, “Lectures on bifurcations in versal families”, Russian Math. Surveys, 27:5 (1972), 54–123  mathnet  crossref  mathscinet  zmath
    2. A. N. Varchenko, “Local topological properties of analytic mappings”, Math. USSR-Izv., 7:4 (1973), 883–917  mathnet  crossref  mathscinet  zmath
    3. A. N. Varchenko, “O rostkakh differentsiruemykh otobrazhenii, topologicheskii tip kotorykh opredelyaetsya konechnoi struei”, UMN, 28:3(171) (1973), 175–176  mathnet  mathscinet  zmath
    4. V. I. Arnol'd, “Remarks on the stationary phase method and Coxeter numders”, Russian Math. Surveys, 28:5 (1973), 19–48  mathnet  crossref  mathscinet  zmath
    5. A. N. Varchenko, “Local topological properties of differentiable mappings”, Math. USSR-Izv., 8:5 (1974), 1033–1082  mathnet  crossref  mathscinet  zmath
    6. A. N. Varchenko, “A theorem on topological versal deformation”, Math. USSR-Izv., 9:2 (1975), 277–296  mathnet  crossref  mathscinet  zmath
    7. V. I. Arnol'd, “Critical points of smooth functions and their normal forms”, Russian Math. Surveys, 30:5 (1975), 1–75  mathnet  crossref  mathscinet  zmath
    8. M.J. Field, “Stratifications of equivariant varieties”, BAZ, 16:2 (1977), 279  crossref
    9. K. K. Karchyauskas, “A generalized Lefshets theorem”, Funct. Anal. Appl., 11:4 (1977), 311–313  mathnet  crossref  mathscinet  zmath
    10. G. R. Belitskii, “Equivalence and normal forms of germs of smooth mappings”, Russian Math. Surveys, 33:1 (1978), 107–177  mathnet  crossref  mathscinet  zmath
    11. A. N. Varchenko, S. M. Gusein-Zade, “The topology of caustics, wave fronts, and degeneracies of critical points”, Russian Math. Surveys, 39:2 (1984), 209–210  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. A. N. Varchenko, “Estimate of the number of zeros of an abelian integral depending on a parameter and limit cycles”, Funct. Anal. Appl., 18:2 (1984), 98–108  mathnet  crossref  mathscinet  zmath  isi
    13. V. A. Vassiliev, “Sharpness and the local Petrovskii condition for strictly hyperbolic operators with constant coefficients”, Math. USSR-Izv., 28:2 (1987), 233–273  mathnet  crossref  mathscinet  zmath
    14. V. P. Leksin, “Meromorphic Pfaffian systems on complex projective spaces”, Math. USSR-Sb., 57:1 (1987), 211–227  mathnet  crossref  mathscinet  zmath
    15. A. I. Degtyarev, V. M. Kharlamov, “Topological properties of real algebraic varieties: du coté de chez Rokhlin”, Russian Math. Surveys, 55:4 (2000), 735–814  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. Sabir M. Gusein-Zade, Ignacio Luengo, Alejandro Melle-Hernández, “On the zeta-function of a polynomial at infinity”, Bulletin des Sciences Mathématiques, 124:3 (2000), 213  crossref  elib
    17. W. Ebeling, S. M. Gusein-Zade, “Radial Index and Euler Obstruction of a 1-Form on a Singular Variety”, Geom Dedicata, 113:1 (2005), 231  crossref  mathscinet  zmath  isi  elib
    18. S. M. Gusein-Zade, D. Siersma, “Deformations of polynomials and their zeta-functions”, Journal of Mathematical Sciences (New York), 144:1 (2007), 3782  crossref  mathscinet
    19. Daniel Daigle, “On polynomials in three variables annihilated by two locally nilpotent derivations”, Journal of Algebra, 310:1 (2007), 303  crossref
    20. S. M. Gusein-Zade, “Integration with respect to the Euler characteristic and its applications”, Russian Math. Surveys, 65:3 (2010), 399–432  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. G. G. Gusev, “Zeta-function of a polynomial on a complete intersection and Newton polytopes”, St. Petersburg Math. J., 23:3 (2012), 511–519  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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