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Lê's vanishing polyhedron for a family of mixed functions

We study real analytic isolated singularities of type f:(Cn+m+1,0)→(C,0) with f(z,w)=g(z)+∑i=1m+1hi(wi,w¯i) , where g is holomorphic and each hi is a mixed polynomial, with z=(z1,⋯,zn) and w=(w1,⋯,wm+1) . We construct a Lê's vanishing polyhedron for f , which describes the degeneration of its Milnor fiber Ff to the singular fiber. Then we prove that Ff is homotopy equivalent to the join Fg∗Fh1∗⋯∗Fhm+1 , where Fg is the Milnor fiber of g and Fhi is the Milnor fiber of hi . This implies that Ff has the homotopy type of a bouquet of spheres Sn+m . So we can define the Milnor number μ(f) as the number of spheres in that bouquet, as in the complex setting.