# zbMATH — the first resource for mathematics

On a singular variety associated to a polynomial mapping. (English) Zbl 1293.32024
Summary: In the paper [“Geometry of polynomial mapping at infinity via intersection homology”, Ann. Inst. Fourier (to appear)], the second author and the third author associated to a given polynomial mapping $$F: \mathbb C^2 \to \mathbb C^2$$ with nonvanishing Jacobian a variety whose homology or intersection homology describes the geometry of singularities at infinity of the mapping. We generalize that result.
##### MSC:
 32H35 Proper holomorphic mappings, finiteness theorems
##### Keywords:
polynomial mapping; Jacobian; intersection homology
Full Text:
##### References:
 [1] J. Bochnak, M. Coste and M.-F. Roy, G\'eom\'etrie alg\'ebrique r\'eelle, Springer 1987. [2] J-P. Brasselet, Introduction to intersection homology and perverse sheaves, to appear. [3] M.GoreskyandR.MacPherson,Intersectionhomology.II.Invent.Math.72(1983),77-129. DOI: 10.1007/BF01389130 [4] M.GoreskyandR.MacPherson,Intersectionhomologytheory,Topology19(1980),135-162. DOI: 10.1016/0040-9383(80)90003-8 [5] R. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings. Amer. J. Math. 102 (1980), no. 2, 291302. · Zbl 0465.14012 [6] Z. Jelonek, The set of point at which polynomial map is not proper, Ann. Polon. Math. 58 (1993), no. 3, 259-266. · Zbl 0806.14009 [7] Z. Jelonek, Testing sets for properness of polynomial mappings, Math. Ann. 315 (1999), no. 1, 1-35. · Zbl 0946.14039 [8] Z. Jelonek, Geometry of real polynomial mappings, Math. Z. 239 (2002), no. 2, 321-333. · Zbl 0997.14017 [9] O.H. Keller, Ganze Cremonatransformationen Monatschr. Math. Phys. 47 (1939) pp. 229-306. [10] T. Mostowski, Some properties of the ring of Nash functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 2, 245-266. · Zbl 0335.14001 [11] T. B. T. Nguyen, \'Etude de certains ensembles singuliers associ\'es ‘a une application polynomiale, Thesis, In preparation. [12] T. B. T. Nguyen, Stratifications de la vari\'et\'e asymptotique associ\'ee ‘a une application polynomiale, In preparation. [13] S. Pinchuk, A counterexample to the strong Jacobian conjecture, Math. Zeitschrift, 217, 1-4, (1994). DOI: 10.1007/BF02571929 · Zbl 0874.26008 [14] A. Valette et G. Valette, Geometry of polynomial mappings at infinity via intersection homology, to appear in Ann. Inst. Fourier. · Zbl 1396.14050 [15] G. Valette, L\inftyhomology is an intersection homology, Adv. Math. 231 (2012), no. 3-4, 1818-1842. · Zbl 1258.14024 [16] H. Whitney, Tangents to an analytic variety, Ann of Math, Second Series, Vol. 81, No. 3, pp. 496-549. · Zbl 0152.27701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.