On the geometry of polynomial mappings at infinity
[Sur la géométrie à l’infini des applications polynomiales]
Valette, Anna ; Valette, Guillaume
Annales de l'Institut Fourier, Tome 64 (2014), p. 2147-2163 / Harvested from Numdam
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On associe à une application polynomiale de ℂ 2 dans lui-même à Jacobien constant non nul, une variété dont l’homologie ou l’homologie d’intersection décrit la géométrie à l’infini de cette application.

We associate to a given polynomial map from ℂ 2 to itself with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2907
Classification:  14P10,  14R15,  32S20,  55N33
Mots clés: singularités à l’infini, valeurs asymptotiques, homologie d’intersection, conjecture Jacobienne. 
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     author = {Valette, Anna and Valette, Guillaume},
     title = {On the geometry of polynomial mappings at infinity},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {2147-2163},
     doi = {10.5802/aif.2907},
     zbl = {06387334},
     mrnumber = {3330934},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_5_2147_0}
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Valette, Anna; Valette, Guillaume. On the geometry of polynomial mappings at infinity. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2147-2163. doi : 10.5802/aif.2907. http://gdmltest.u-ga.fr/item/AIF_2014__64_5_2147_0/

[1] Bochnak, J.; M., Coste; Roy, M.-F. Géométrie algébrique réelle, Springer (1987) | MR 949442 | Zbl 0633.14016

[2] Goresky, M.; Macpherson, R. Intersection homology theory, Topology, Tome 19 (1980) no. 2, pp. 135-162 | Article | MR 572580 | Zbl 0448.55004

[3] Goresky, M.; Macpherson, R. Intersection homology. II, Invent. Math., Tome 72 (1983), pp. 77-129 | Article | MR 696691 | Zbl 0529.55007

[4] Hardt, R. Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math., Tome 102 (1980) no. 2, pp. 291-302 | Article | MR 564475 | Zbl 0465.14012

[5] Jelonek, Z. The set of point at which polynomial map is not proper, Ann. Polon. Math., Tome 58 (1993) no. 3, pp. 259-266 | MR 1244397 | Zbl 0806.14009

[6] Jelonek, Z. Testing sets for properness of polynomial mappings, Math. Ann., Tome 315 (1999) no. 1, pp. 1-35 | Article | MR 1717542 | Zbl 0946.14039

[7] Jelonek, Z. Geometry of real polynomial mappings, Math. Z., Tome 239 (2002) no. 2, pp. 321-333 | Article | MR 1888227 | Zbl 0997.14017

[8] Keller, O. H. Ganze Cremonatransformationen Monatschr, Math. Phys., Tome 47 (1939), pp. 229-306 | Zbl 0021.15303

[9] Kirwan, F.; Woolf, J. An Introduction to Intersection Homology Theory, Chapman & Hall/CRC (2006) | MR 2207421 | Zbl 1106.55001

[10] Mostowski, T. Some properties of the ring of Nash functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Tome 3 (1976) no. 2, pp. 245-266 | Numdam | MR 412180 | Zbl 0335.14001

[11] Valette, G. L ∞ homology is an intersection homology, Adv. in Math., Tome 231 (2012) no. 3-4, pp. 1818-1842 | Article | MR 2964625 | Zbl 1258.14024