it is a question of determining by algebraic methods the coefficients of the sum.
In the weighted homogeneous case this can be carried out with Macaulay-
Bezout numbers. Using them, we deduce the formulas obtained by Greuel-Hamm
[GrH] and Giusti [Gi] for weighted homogeneous ICIS's. We also compute singular
Milnor numbers for almost free arrangments on complete intersections, and for the
fibers of singular projections of discriminants.
An alternate way to compute singular Milnor numbers is obtained by an
analogue of the Le-Greuel formula for relative Milnor fibers. We again obtain a
formula for almost free complete intersections in terms of the length of a
In Part IV, we complete the calculation of "vanishing Euler characteristics"
for the pullbacks of almost free divisors and complete intersections. In part I, the
computation is given in the case "n p". If "n p" then the pullback need no
longer be almost free. Nonetheless, theorem 3 gives a general composition formula
for the "vanishing Euler characteristics" in the case "n p". This is applied to
obtain formulas for the Euler characteristic of nonisolated hypersurface singularities
defined via a composition of a germ defining an ICIS with a plane curve singularity;
these are the "generalized Zariski examples". Nemethi [N] and Massey-Siersma
[MS] obtained formulas involving specific intersection invariants of the curve
singularity. We give an alternative general composition formula which involves
only the 3 main "Milnor numbers" of the various objects, which can be
algebraically computed. Moreover, this composition formula also applies to other
situations such as pullbacks of almost free divisors by finite map germs; when
applied to modified Zariski examples it provides examples of nonREALizable
singular complex cycles, which also occur for bifurcation questions in §11.
In the last section, we consider the bifurcation problems for unfoldings of
isolated hypersurface singularities of finite bifurcation codimension and compute
the singular Milnor numbers for the bifurcation set. We show that there are
universally nonREALizable vanishing cycles for bifurcation problems.