## Milnor fibration at infinity.(English)Zbl 0806.57021

Suppose $$f: C^ n\to C$$ is a complex polynomial. The authors show that under certain conditions there is a Milnor fibration at infinity, i.e., if $$S_ R$$ is a sphere around the origin of sufficiently large radius then the map $$f/ | f|: S_ R- f^{-1}(0)\to S^ 1$$ is a fibration. They also prove a stability theorem in certain circumstances.

### MSC:

 57R99 Differential topology 14B05 Singularities in algebraic geometry

### Keywords:

complex polynomial; Milnor fibration at infinity
Full Text:

### References:

 [1] Broughton, S. A., On the topology of polynomial hypersurfaces, Proceedings of Symposia in Pure Mathematics, Volume 40 (1983), Part 1 · Zbl 0526.14010 [2] Broughton, S. A., Milnor numbers and the topology of polynomial hypersurfaces, Invent. math., 92, 217-241 (1988) · Zbl 0658.32005 [3] Eisenbud, D.; Neumann, W., Three-dimensional link theory and invariants of plane curve singularities, (Ann. of Math. Studies, 101 (1985), Princeton Univ. Press) · Zbl 0628.57002 [4] Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. math., 32, 1-31 (1976) · Zbl 0328.32007 [5] Milnor, J., Morse Theory, (Ann. of Math. Studies, 51 (1963), Princeton Univ. Press) [6] Milnor, J., Singular Points of Complex Hypersurfaces, (Ann. of Math. Studies, 61 (1968), Princeton Univ. Press) · Zbl 0184.48405 [7] Némethi, A., Théorie de Lefschetz pour les variétés algébriques affines, C.R. Acad. Sc. Paris (1986), t. 303, Serie I, n.12 · Zbl 0612.14007 [8] Némethi, A., Lefschetz theory for complex affine varieties, Rev. Roum. Math. Pures Appl., 33, 233-260 (1988) · Zbl 0665.14003 [9] Némethi, A.; Zaharia, A., On the Bifurcation Set of a Polynomial Function and Newton Boundary, Publ. RIMS Kyoto Univ., 26, 681-689 (1990) · Zbl 0736.32024 [10] Neumann, W. D., Complex algebraic plane curves via their links at infinity, Invent. math., 98, 445-489 (1989) · Zbl 0734.57011 [11] Neumann, W. D.; Rudolf, L., Corrigendum: Unfolding in knot theory, Math. Ann., 282, 349-351 (1988) · Zbl 0675.57011 [12] Oka, M., On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math. Soc. Japan, 31, 435-450 (1979) · Zbl 0408.35012 [13] Oka, M., On the topology of the Newton boundary II, J. Math. Soc. Japan, 32, 65-92 (1980) · Zbl 0417.14004 [14] Oka, M., On the topology of the Newton boundary III, J. Math. Soc. Japan, 34, 541-549 (1982) · Zbl 0476.32016 [15] Pham, F., La descente des cols par les onglets de Lefschetz, avec vues sur Gauss-Manin, Systèmes différentiels et singularités, 130 (Juin-Juillet 1983), Astérisque [16] Varchenko, A. N., Theorems on Topological Equisingularity of Families of Algebraic Varieties and Families of Polynomial Maps, Izvestiya Akad. Nauk, 36, 957-1019 (1972) [17] Hà, H. V.; Lê, D. T., Sur la topologie des polynômes complexes, Acta Math. Vietnamica, 9, 21-32 (1984) · Zbl 0597.32005
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