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Milnor numbers and the topology of polynomial hypersurfaces. (English) Zbl 0658.32005

Let F: \({\mathbb{C}}^{n+1}\to {\mathbb{C}}\) be a polynomial. F is called tame if \(\| \partial F(x)\| >a\) for \(\| x\| >R\), where a and R are some positive real numbers. In this case, F has only a finite number of critical points. Let \(\mu\) (F) (resp. \(\mu^ c(F))\) be the sum of the Milnor numbers at all critical points of F in \({\mathbb{C}}^{n+1}\) (resp. lying on \(F^{-1}(c))\). Then the following theorem holds: For any \(c\in {\mathbb{C}}\), \(F^{-1}(c)\) has the homotopy type of a bouquet of \(\mu (F)- \mu^ c(F)\) spheres of dimension n.
It is also shown that the property for a polynomial to be tame is generic.
In the general case (i.e. without the assumption that F is tame), information on the homology groups of level sets \(F^{-1}(c)\) for generic \(c\in {\mathbb{C}}\) is also obtained in terms of the Milnor numbers of isolated critical points of F.
Reviewer: T.Krasiński

MSC:

32S05 Local complex singularities
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14F45 Topological properties in algebraic geometry
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References:

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