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Bifurcations and topology of meromorphic germs

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New Developments in Singularity Theory

Part of the book series: NATO Science Series ((NAII,volume 21))

Abstract

Maps defined by polynomial functions are traditional objects of interest in algebraic geometry and singularity theory. A polynomial P in n complex variables defines a map P : ℂn → ℂ. The map P is not a locally trivial flbration over critical values of P. However, since the source ℂn is not compact, the map P fails to be a locally trivial fibration over some other values as well. It is well known that a polynomial map defines a locally trivial fibration over the complement to a finite set in ℂ (the bifurcation set of P): [41, 45, 47].

Partially supported by RFBR 98-01-00612, NWO-RFBR 047.008.005.

Partially supported by DGCYT PB97-0284-C02-01.

Partially supported by DGCYT PB97-0284-C02-01.

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References

  1. N. A’Campo, La fonction zeta d’une monodromie, Comment. Math. Helv. 50 (1975) 233–248.

    Article  MathSciNet  Google Scholar 

  2. V.I. Arnold, Singularities of fractions and behaviour of polynomials at infinity, Proc. Steklov Math. Inst. 221 (1998) 48–68.

    Google Scholar 

  3. V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of Differentiable Maps, Vol. II, Birkhäuser, Boston-Basel-Berlin, 1988.

    Google Scholar 

  4. E. Artal Bartolo, P. Cassou-Nogües and A. Dimca, Sur la topologie des polynAômes complexes, Singularities: the Brieskorn anniversary volume (eds V.I. Arnold, G.-M. Greuel and J.H.M. Steenbrink), Progress in Math. 162 (Birkhäuser, 1998) pp 317–343.

    Google Scholar 

  5. E. Artal Bartolo, I. Luengo and A. Melle Hernández, Milnor number at infinity, topology and Newton boundary of a polynomial function, Math. Zeits. 233:4 (2000) 679–696.

    Article  MATH  Google Scholar 

  6. E. Artal Bartolo, I. Luengo and A. Melle Hernández, On the topology of a generic fibre of a polynomial function, Comm. Algebra 28:4 (2000) 1767–1787.

    Article  MathSciNet  Google Scholar 

  7. D.N. Bernshtein, The number of roots of a system of equations, Funct. Anal. Appl. 9:3 (1975) 1–4.

    Google Scholar 

  8. S.A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988) 217–241.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Cassou-Nogues and A. Dimca, Topology of complex polynomials via polar curves, Kodai Math. J. 22 (1999) 131–139.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Dimca, On the connectivity of complex affine hypersurfaces, Topology 29 (1990) 511–514.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Dimca, On the Milnor fibrations of weighted homogeneous polynomials, Compositio Math. 76 (1990) 19–47.

    MathSciNet  MATH  Google Scholar 

  12. A. Dimca, Monodromy and Hodge theory of regular functions, this volume.

    Google Scholar 

  13. A. Dimca and A. Neméthi, On the monodromy of complex polynomials, preprint math.AG/9912072, Duke Math. Jour., to appear.

    Google Scholar 

  14. R. García López, Exponential sums and singular hypersurfaces, Manuscripta Math. 97 (1998) 45–58.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. García López and A. Neméthi, On the monodromy at infinity of a polynomial map I, Compositio Math. 100 (1996) 205–231.

    MathSciNet  MATH  Google Scholar 

  16. S.M. Gusein-Zade, On a problem of B. Teissier, Topics in singularity theory. V.I. Arnold’s 60th anniversary collection (eds A. Khovanskii, A. Varchenko and V. Vassiliev), A.M.S. Translations, Ser. 2, 180, (Amer. Math. Soc, Providence, RI, 1997) pp 117–125.

    Google Scholar 

  17. S.M. Gusein-Zade, I. Luengo and A. Melle Hernández, Partial resolutions and the zeta-function of a singularity, Comment. Math. Helv. 72:2 (1997) 244–256.

    Article  MathSciNet  MATH  Google Scholar 

  18. S.M. Gusein-Zade, I. Luengo and A. Melle Hernández, Zeta-functions for germs of meromorphic functions and Newton diagrams, Funct. Anal. Appl. 32:2 (1998) 26–35.

    Article  MathSciNet  Google Scholar 

  19. S.M. Gusein-Zade, I. Luengo and A. Melle Hernández, On atypical values and local monodromies of meromorphic functions, Tr. Mat. Inst. Steklova (Proc. Steklov Math. Inst.) 225 (1999) (Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 60th birthday) 168–176.

    Google Scholar 

  20. S.M. Gusein-Zade, I. Luengo and A. Melle Hernández, On the topology of germs of meromorphic functions, Algebra i Analiz 11:5 (1999) 92–99; translation in: St. Petersburg Math. Journal 11:5, 775-780.

    Google Scholar 

  21. S.M. Gusein-Zade, I. Luengo and A. Melle Hernández, On the zeta-function of a polynomial at infinity, Bull. Sci. Math. 124:3 (2000) 213–224.

    Article  MathSciNet  MATH  Google Scholar 

  22. Há Huy Vui and Lê Düng Tràng, Sur la topologie des polynAômes complexes, Acta Math. Vietnam. 9:5 (1984) 21–32 (1985).

    MathSciNet  MATH  Google Scholar 

  23. A.G. Khovanskii, Newton polyhedra and toroidal varieties, Funct. Anal. Appl. 11:4 (1977) 56–64.

    Google Scholar 

  24. D.T. Lê and C.P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976) 67–78.

    Article  MathSciNet  MATH  Google Scholar 

  25. D.T. Lê and Claude Weber, Equisingularité dans les pinceaux de germes de courbes planes et C 0suffisance, l’Enseignement Math. 43 (1997) 355–380.

    MATH  Google Scholar 

  26. A. Libgober and S. Sperber, On the zeta-function of monodromy of a polynomial map, Compositio Math. 95 (1995) 287–307.

    MathSciNet  MATH  Google Scholar 

  27. A. Némethi, Global Sebastiani-Thom theorem for polynomial maps, J. Math. Soc. Japan 43 (1991) 213–218.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Némethi and A. Zaharia, Milnor fibration at infinity, Indag. Math, (new ser.) 3 (1992) 323–335.

    Article  MATH  Google Scholar 

  29. M. Oka, Principal zeta-function of non-degenerate complete intersection singularity, J. Fac. Sci. Uni. Tokyo Sect. IA 37 (1990) 11–32.

    MathSciNet  MATH  Google Scholar 

  30. A. Parusiński, On the bifurcation set of a complex polynomial with isolated singularities at infinity, Compositio Math. 97 (1995) 369–338.

    MathSciNet  MATH  Google Scholar 

  31. A. Parusiński, Topological triviality of μ-constant deformations of type f(x) + tg(x), Bull. London Math. Soc. 31 (1999) 686–692.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Parusiński and P Pragacz, A formula for the Euler characteristic of singular hypersurfaces, J. Alg. Geom. 4 (1995) 337–351.

    MATH  Google Scholar 

  33. A. Parusiński and P Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Algebraic Geom. 10 (2001), 63–79.

    MathSciNet  MATH  Google Scholar 

  34. C. Sabbah, Hypergeometric periods for a tame polynomial, Comptes Rendus Acad. Sci. Paris Ser. I 328 (1999) 603–608.

    MathSciNet  MATH  Google Scholar 

  35. D. Siersma, The monodromy of a series of hypersurface singularities, Comment. Math. Helvet. 65 (1990) 181–197.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Siersma, The vanishing topology of non-isolated singularities, this volume.

    Google Scholar 

  37. D. Siersma and M. Tibăr, Singularities at infinity and their vanishing cycles, Duke Math. J. 80 (1995) 771–783.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Siersma and M. Tibăr, Topology of polynomial functions and monodromy dynamics, Comptes Rendus Acad. Sci. Paris Ser. I 327 (1998) 655–660.

    MATH  Google Scholar 

  39. D. Siersma and M. Tibăr, Vanishing cycles and singularities of meromorphic functions, preprint math.AG/9905108.

    Google Scholar 

  40. D. Siersma and M. Tibăr, Singularities at infinity and their vanishing cycles, II. Monodromy, preprint 1088 (Jan. 1999), Utrecht University. http://www.mi.aau.dk/esn/abstracts.html/144/144.html.

  41. R. Thom, Ensembles et morphismes stratifiées, Bull. Amer. Math. Soc. 75 (1969) 240–284.

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Tibăr, Topology at infinity of polynomial mappings and Thom regularity condition, Compositio Math. 111 (1998) 89–109.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Tibär, Regularity at infinity of real and complex polynomial functions, Singularity Theory (eds Bill Bruce and David Mond), London Math. Soc. Lecture Notes 263 (Cambridge University Press. 1999) pp 249–264.

    Google Scholar 

  44. M. Tibăr, On the monodromy fibration of polynomial functions with singularities at infinity, Comptes Rendus Acad. Sci. Paris Ser. I 324 (1997) 1031–1035.

    MATH  Google Scholar 

  45. A.N. Varchenko, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Math USSR Izvestija 6 (1972) 949–1008.

    Article  Google Scholar 

  46. A.N. Varchenko, Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37 (1976) 253–262.

    Article  MathSciNet  MATH  Google Scholar 

  47. J.L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976) 295–312.

    Article  MathSciNet  MATH  Google Scholar 

  48. O.Y. Viro, Some integral calculus based on Euler characteristic, Topology and Geometry — Rohlin seminar Springer Lecture Notes in Math. 1346 127–138.

    Google Scholar 

  49. Y.N. Yomdin, Complex surfaces with a one-dimensional set of singularities, Siberian Math. J. 5 (1975) 748–762.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Moscow State University, Russia

    Sabir Gusein-Zade

  2. Departamento de Álgebra, Universidad Complutense de Madrid, Germany

    Ignacio Luengo & Alejandro Melle Hernández

Authors
  1. Sabir Gusein-Zade
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  2. Ignacio Luengo
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  3. Alejandro Melle Hernández
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University Utrecht, Utrecht, The Netherlands

    D. Siersma

  2. Mathematical Sciences, Liverpool, UK

    C. T. C. Wall

  3. Moscow Aviation Institute, Moscow, Russian Republic

    V. Zakalyukin

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Gusein-Zade, S., Luengo, I., Hernández, A.M. (2001). Bifurcations and topology of meromorphic germs. In: Siersma, D., Wall, C.T.C., Zakalyukin, V. (eds) New Developments in Singularity Theory. NATO Science Series, vol 21. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0834-1_12

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  • DOI: https://doi.org/10.1007/978-94-010-0834-1_12

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