Milnor numbers and the topology of polynomial hypersurfaces

Summary LetF: n + 1 be a polynomial. The problem of determining the homology groupsH_q(F^–1(c)), c , in terms of the critical points ofF is considered. In the best case it is shown, for a certain generic class of polynomials (tame polynomials), that for allc,F^–1(c) has the homotopy type of a bouquet of -^cn-spheres. Here is the sum of all the Milnor numbers ofF at critical points ofF and ^c is the corresponding sum for critical points lying onF^–1(c). A second best case is also discussed and the homology groupsH_q(F^–1(c)) are calculated for genericc. This case gives an example in which the critical points at infinity ofF must be considered in order to determine the homology groupsH_q(F^–1(c)).