Artal Bartolo, Enrique and Luengo Velasco, Ignacio and Melle Hernández, Alejandro (2000) On the topology of a generic fibre of a polynomial function. Communications in Algebra, 28 (4). pp. 1767-1787. ISSN 0092-7872
In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.
|Uncontrolled Keywords:||Complex polynomial function, atypical values at infinity|
|Subjects:||Sciences > Mathematics > Algebra|
E. Artal Bartolo, Forme de Jordan de la monodromie des singularitées superisolées de surfaces, Mem. Amer. Math. Soc., vol.109, no. 525, American Mathematical Society, Providence RI, 1994.
E. Artal, I. Luengo, A. Melle, Milnor number at infinity, topology and Newton boundary of a polynomial function, Preprint (1997).
S.A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. math. 92 (1988), 217–241.
A. Dimca, On the connectivity of a complex affine hyperserfaces, Topology 29 (1990), 511–514.
A. Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992.
R. García López, A. Némethi, On the monodromy at infinity of a polynomial map, Compositio Math. 100 (1996), 205–231. H.V. Hà, D.T. Lê, Sur la topologie des polynômes complexes, Acta Math. Vietnamica 9 (1984), 21–32.
A. Melle-Hernández, Milnor numbers of surface singularities, Israel J. Math. (1998) (to appear). cf.
A. Némethi, Théorie de Lefschetz pour les variétés algébriques affines, C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 567–570.
A. Némethi, Global Sebastiani-Thom theorem for polynomial maps, J. Math. Soc. Japan 43 (1991), 213–218.
A. Némethi, A. Zaharia, On the bifurcation set of a polynomial function and Newton boundary, Publ. RIMS Kyoto Univ. 26 (1990), 681–689.
A. Parusiński, A generalization of the Milnor number, Math. Ann. 281 (1988), 247–254.
A. Parusiński, Multiplicity of the dual variety, Bull. London Math. Soc. 23 (1991), 428–436.
A. Parusiński, On the bifurcation set of a complex polynomial with isolated singularities at the infinity, Compositio Math. 97 (1995), 369–384.
A. Parusiński, P. Pragacz, A Formula for the Euler characteristic of singular hypersurfaces, J. Algebraic Geometry 4 (1995), 337–351.
F. Pham, Vanishing homologies and the n variables saddlepoint method, Proc. A.M.S. Symp. in Pure Math., vol. 40, 1983, pp. 319–335.
D. Siersma, M. Tibǎr, Singularities at infinity and their vanishing cycles, Duke Math. J. 80 (1995), 771–783.
G.M. Greuel, G. Pfister, H. Schoenemann, SINGULAR. A computer algebra system for singularity theory and algebraic geometry, It is available via anonymous ftp from helios.mathematik.uni-kl.de.
J.L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Inv. math. 36 (1976), 295–312.
|Deposited On:||18 Oct 2012 12:01|
|Last Modified:||12 Apr 2013 09:22|
Repository Staff Only: item control page