Comptes Rendus
no. 12
Non-reality and non-connectivity of complex polynomials
[Non réalité et non connectivité des polynômes complexes]
Arnaud Bodin1
1 Laboratoire Agat, UFR de mathématiques, Université Lille 1, 59655 Villeneuve d'Ascq cedex, France
Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1039-1042.

Pour les polynômes de deux variables complexes, nous construisons des contre-exemples aux questions suivantes : à équivalence topologique près, peut-on toujours trouver une équation réelle à un polynôme complexe (Lee Rudolph) ? Deux polynômes topologiquement équivalents peuvent-ils être reliés par une famille de polynômes topologiquement équivalents ?

Using the same method we provide negative answers to the following questions: is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph)? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomials?

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02597-9

Arnaud Bodin 1

1 Laboratoire Agat, UFR de mathématiques, Université Lille 1, 59655 Villeneuve d'Ascq cedex, France 
@article{CRMATH_2002__335_12_1039_0,
     author = {Arnaud Bodin},
     title = {Non-reality and non-connectivity of complex polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1039--1042},
     publisher = {Elsevier},
     volume = {335},
     number = {12},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02597-9},
     language = {en},
}
TY  - JOUR
AU  - Arnaud Bodin
TI  - Non-reality and non-connectivity of complex polynomials
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 1039
EP  - 1042
VL  - 335
IS  - 12
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02597-9
LA  - en
ID  - CRMATH_2002__335_12_1039_0
ER  - 
%0 Journal Article
%A Arnaud Bodin
%T Non-reality and non-connectivity of complex polynomials
%J Comptes Rendus. Mathématique
%D 2002
%P 1039-1042
%V 335
%N 12
%I Elsevier
%R 10.1016/S1631-073X(02)02597-9
%G en
%F CRMATH_2002__335_12_1039_0
Arnaud Bodin. Non-reality and non-connectivity of complex polynomials. Comptes Rendus. Mathématique, Volume 335 (2002) no. 12, pp. 1039-1042. doi : 10.1016/S1631-073X(02)02597-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02597-9/

[1] N. A'Campo Real deformations and complex topology of plane curve singularities, Ann. Fac. Sci. Toulouse Math., Volume 8 (1999), pp. 5-23

[2] E. Artal, J. Carmona, J. Cogolludo, Effective invariants of braid monodromy and topology of plane curves, Preprint

[3] H. Bass; E. Connell; D. Wright The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., Volume 7 (1982), pp. 287-330

[4] A. Bodin Classification of polynomials from ℂ 2 to ℂ with one critical value, Math. Z., Volume 242 (2002), pp. 303-322

[5] A. Bodin, Invariance of Milnor numbers and topology of complex polynomials, Comment. Math. Helv., to appear

[6] A. Bodin, Computation of Milnor numbers and critical values in affine space and at infinity, Preprint

[7] D. Eisenbud; W. Neumann Three-Dimensional Link Theory and Invariants of Plane Curve Singularities, Ann. Math. Stud., 110, Princeton University Press, 1985

[8] G.-M. Greuel, G. Pfister, H. Schönemann, Singular 2.0: a computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern, 2001

[9] V. Kharlamov; V. Kulikov Diffeomorphisms, isotopies, and braid monodromy factorizations of plane cuspidal curves, C. R. Acad. Sci. Paris, Série I, Volume 333 (2001), pp. 855-859

[10] L. Rudolph, Private communication, Geneva, 1998

Cité par Sources :

Commentaires - Politique