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Sur la topologie des polynômes complexes

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Singularities

Part of the book series: Progress in Mathematics ((PM,volume 162))

Résumé

Soit f ∊ ℂ[x i,... ,x n ] un polynôme de degré d. On sait [P], [ST] qu’il existe un ensemble fini A C C tel que

$$f:{C^n}\backslash {f^{ - 1}}(\Lambda ) \to C\backslash \Lambda $$

est une fibration localement triviale. Dorénavant A désigne le plus petit ensemble qui possède cette propriété. Si t ∊ Λ, la fibre F t = f −1(t) est appellée fibre irrégulière de f, sinon elle est dite régulière ou générique. Soit δ un nombre réel positif assez petit, δ ∉ Λ. On note

$${D_\delta } = \{ t \in C|\left| t \right| < \delta \} ,{S_\delta } = \partial {{\bar D}_\delta }et{T_\delta } = {f^{ - 1}}({D_\delta })$$

Si la fibre F 0 est régulière, alors on a les deux faits suivants.

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References

  1. S. Abhyankar; T. Moh: Embeddings of the line in the plane. J. Reine Angew. Math. 267, 148–166, (1975).

    MathSciNet  Google Scholar 

  2. N. A’Campo: La fonction Zêta d’une monodromie. Comment. Math. Helvetici 50, 233–248, (1975).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Artal Bartolo: Une démonstration géométrique du théorème d’ Abyankar-Moh. J. Reine angew. Math 464, 97–108, (1995).

    MathSciNet  MATH  Google Scholar 

  4. E. Artal Bartolo; P. Cassou-Noguès; I. Luengo Velasco: On polynomials whose fibers are irreducible with no critical points. Math. Ann. 299, 477–490, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Brieskorn; H. Knörrer: Plane Algebraic Curves. Birkhäuser, Boston, (1986).

    MATH  Google Scholar 

  6. S.A. Broughton: Milnor number and the topology of polynomial hypersur-faces. Invent. Math. 92, 217–241, (1988).

    Article  MathSciNet  MATH  Google Scholar 

  7. C.H. Clemens: Degeneration of Kähler manifolds. Duke Math. J. 44, 215–290, (1977).

    Article  MathSciNet  MATH  Google Scholar 

  8. A.D.R. Choudary; A. Dimca: Complex hypersurfaces diffeomorphic to affine spaces. Kodai Math. J. 17, 171–178, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Deligne: Théorie de Hodge II. Publ. Math. I.H.E.S. 40, 5–58, (1971).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Dimca: Singularities and topology of hypersurfaces. Universitext, Springer, (1992).

    Book  MATH  Google Scholar 

  12. R. García López; A. Nemethi: On the monodromy at infinity of a polynomial map, I. Compositio Math. 100, 205–231, (1996).

    MathSciNet  MATH  Google Scholar 

  13. Ha Huy Vui; A. Zaharia: Families of polynomials with total Milnor number constant. Math. Ann. 304, 481–488, (1996).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Fourrier: Topologie d’un polynôme de deux variables complexes au voisinage de l’infini. Thèse Université de Toulouse, (1993).

    Google Scholar 

  15. W. Fulton: Introduction to toric varieties Annals of Math. Studies 131. Princeton Univ. Press, (1993).

    Google Scholar 

  16. H. Hamm: Zum Homotopietyp Steinscher Raeume. J. reine angew. Math. 338, 121–135, (1983).

    MathSciNet  MATH  Google Scholar 

  17. A.J. de Jong; J.H.M. Steenbrink: Proof of a conjecture of W. Veys. Indag. Math. N. S. 6, 99–104, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Kaliman: Two remarks on polynomials in two variables. Pacific J. Math. 154, 285–295, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  19. Le Dung Trang; C. Weber: Polynômes à fibres rationnelles et conjecture jacobienne à 2 variables C. R. Acad. Sci. Paris, t. 320, 581–584, (1995).

    MATH  Google Scholar 

  20. S. Lojasiewicz: Triangulation of semi-analytic sets. Ann. Sc. Norm. Sup. Pisa 18, 449–474, (1964).

    MathSciNet  MATH  Google Scholar 

  21. J. Milnor: Singular Points of Complex Hypersurfaces. Annals of Math. Studies 61, Princeton, (1968).

    Google Scholar 

  22. M. Miyanishi; T. Sugie: Generically rational polynomials. Osaka J. Math. 17, 339–362, (1980).

    MathSciNet  MATH  Google Scholar 

  23. D. Mumford: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. I.H.E.S. 9, 5–22, (1961).

    MathSciNet  MATH  Google Scholar 

  24. A. Nemethi; A. Zaharia: Milnor fibration at infinity. Indag. Math. 3, 323–335, (1992).

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Pham: Vanishing homologies and the n variable saddlepoint method. Proc. Symp. Pure Math. 40, 319–333, (1983).

    MathSciNet  Google Scholar 

  26. A. Parusinski: A note on singularities at infinity of complex polynomials. Banach Center Math. Publ.

    Google Scholar 

  27. J.P. Serre: Corps locaux. Hermann, Paris, (1968).

    Google Scholar 

  28. D. Siersma; M. Tibar: Singularities at infinity and their vanishing cycles. Duke Math. J. 80, 771–783, (1995).

    Article  MathSciNet  MATH  Google Scholar 

  29. E.H. Spanier: Algebraic Topology. Mc Graw Hill, (1966).

    Google Scholar 

  30. O. Zariski: Algebraic Surfaces. Chelsea Publishing Press, (1948).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Zaragoza, 50009, Zaragoza, Espagne

    Enrique Artal-Bartolo

  2. Laboratoire de mathématiques pures, Université Bordeaux I, 350 Cours de la Libération, 33405, Talence, France

    Pierrette Cassou-Noguès & Alexandru Dimca

Authors
  1. Enrique Artal-Bartolo
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  2. Pierrette Cassou-Noguès
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  3. Alexandru Dimca
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Editor information

Editors and Affiliations

  1. Department of Geometry and Topology, Steklov Mathematical Institute, 8, Gubkina Stree, 117966, Moscow GSP-1, Russia

    V. I. Arnold

  2. CEREMADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, Postfach 3049, F-75775, Paris Cedex 16e, France

    V. I. Arnold

  3. Fachbereich Mathematik, Universität Kaiserslautern, D-67653, Kaiserslautern, Germany

    G.-M. Greuel

  4. Subfaculteit Wiskunde, Katholieke Universiteit Nijmegen Toernooiveld, NL-6525 ED, Nijmegen, The Netherlands

    J. H. M. Steenbrink

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Artal-Bartolo, E., Cassou-Noguès, P., Dimca, A. (1998). Sur la topologie des polynômes complexes. In: Arnold, V.I., Greuel, GM., Steenbrink, J.H.M. (eds) Singularities. Progress in Mathematics, vol 162. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8770-0_16

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  • DOI: https://doi.org/10.1007/978-3-0348-8770-0_16

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9767-9

  • Online ISBN: 978-3-0348-8770-0

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