Quotients jacobiens d'applications polynomiales

Artal Bartolo, Enrique; Cassou-Noguès, Philippe; Maugendre, Hélène

Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 399-428.

Soit $\phi :=(f,g):{ℂ}^2\to {ℂ}^2$ où $f$ et $g$ sont des applications polynomiales. Nous établissons le lien qui existe entre le polygone de Newton de la courbe réunion du discriminant et du lieu de non-propreté de $\phi$ et la topologie des entrelacs à l’infini des courbes affines $f^{-1}\left(0\right)$ et $g^{-1}\left(0\right)$. Nous en déduisons alors des conséquences liées à la conjecture du jacobien.

Let $\phi :=(f,g):{ℂ}^2\to {ℂ}^2$ where $f$ and $g$ are polynomial maps. A relationship is established between the following two objects: on the one hand, the Newton polygon of the union of the discriminant curve of $\phi$ and its non-properness locus, and on the other, the topological type of the link at infinity of the affine curves $f^{-1}\left(0\right)$ and $g^{-1}\left(0\right)$. Some consequences related to the Jacobian Conjecture are obtained.

[1] A. Assi Sur l'intersection des courbes méromorphes, C. R. Acad. Sci. Paris, Sér. I Math., Volume 329 (1999) no. 7, pp. 625-628 | DOI | MR | Zbl

[2] P. Cassou-Noguès Diagrams of algebraic curves (2000) (preprint)

[3] E. Artal; P. Cassou-Noguès; A. Dimca Topology of complex polynomials via polar curves, Kodai Math. J., Volume 22 (1999) no. 1, pp. 131-139 | DOI | MR | Zbl

[4] Nguyen Van Chau Non-zero constant Jacobian polynomial maps of ${ℂ}^2$, Ann. Polon. Math., Volume 71 (1999) no. 3, pp. 287-310 | MR | Zbl

[5] D. Eisenbud; W.D. Neumann Three-dimensional link theory and invariance of plane curve singularities, Annals of Mathematics Studies, no 110, Princeton University Press, Princeton NJ, 1985 | Zbl

[6] J. Gwoździewicz; A. PŁoski Formulae for the singularities at infinity of plane algebraic curves, Effective methods in algebraic and analytic geometry, 2000 (Kraków), MR1 886 934 (Univ. Iagel. Acta Math.), Volume 39 (2001), pp. 109-133 | Zbl

[7] W. Jaco Lectures on three-manifold topology, American Mathematical Society, Providence, R.I., 1980 | MR | Zbl

[8] Z. Jelonek Testing sets for properness of polynomial mappings, Math. Ann., Volume 315 (1999) no. 1, pp. 1-35 | DOI | MR | Zbl

[9] D.T. Lê; H. Maugendre; C. Weber Geometry of critical loci, J. London Math. Soc. (2), Volume 63 (2001) no. 3, pp. 533-552 | DOI | MR | Zbl

[10] D.T. Lê; C. Weber A geometrical approach to the Jacobian conjecture for $n=2$, Kodai Math. J., Volume 17 (1994), pp. 374-381 | DOI | MR | Zbl

[11] C. Lescop Global surgery formula for the Casson-Walker invariant, Princeton University Press, Princeton, NJ, 1996 | MR | Zbl

[12] H. Maugendre Discriminant of a germ $\Phi :({ℂ}^2,0)\to ({ℂ}^2,0)$ and Seifert fibred manifolds, J. London Math. Soc. (2), Volume 59 (1999) no. 1, pp. 207-226 | MR | Zbl

[13] T.T. Moh On the Jacobian conjecture and the configurations of roots, J. reine angew. Math., Volume 340 (1983), pp. 140-212 | EuDML | MR | Zbl

[14] W.D. Neumann A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc., Volume 268 (1981) no. 2, pp. 299-344 | DOI | MR | Zbl

[15] W.D. Neumann; P. Norbury Rational polynomials of simple type, Pacific J. Math., Volume 204 (2002) no. 1, pp. 177-207 | DOI | MR | Zbl

[16] P. Russell Good and bad field generators, J. Math. Kyoto Univ., Volume 17 (1977) no. 2, pp. 319-331 | MR | Zbl

[17] B. Teissier Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse, Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972 (Astérisque), Volume no 7-8 (1973), pp. 285-362 | Numdam | Zbl

[18] F. Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I, Invent. Math., Volume 3 (1967), pp. 308-333 | DOI | EuDML | Zbl

[19] F. Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten II, Invent. Math., Volume 4 (1967), pp. 87-117 | DOI | EuDML | MR | Zbl