Milnor numbers and the topology of polynomial hypersurfaces

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Let F: ℂn + 1→ℂ be a polynomial. The problem of determining the homology groups H q (F −1 (c)), c ∈ℂ, in terms of the critical points of F is considered. In the “best case” it is shown, for a certain generic class of polynomials (tame polynomials), that for all c∈ℂ,F −1 (c) has the homotopy type of a bouquet of μ-μ c n-spheres. Here μ is the sum of all the Milnor numbers of F at critical points of F and μ c is the corresponding sum for critical points lying on F −1 (c). A “second best” case is also discussed and the homology groups H q (F −1 (c)) are calculated for genericc∈ℂ. This case gives an example in which the critical points “at infinity” of F must be considered in order to determine the homology groups H q (F −1 (c)).