Lê's vanishing polyhedron for a family of mixed functions

We study real analytic isolated singularities of type f : ( C n + m + 1 , 0 ) → ( C , 0 )with f ( z , w ) = g ( z ) + ∑ i = 1 m + 1h i ( w i , w ¯ i ) , where g is holomorphic and each h i is a mixed polynomial, with z = ( z 1 , ⋯ , z n ) and w = ( w 1 , ⋯ , w m + 1 ) . We construct a Lê's vanishing polyhedron for f , which describes the degeneration of its Milnor fiber F f to the singular fiber. Then we prove that F f is homotopy equivalent to the joinF g ∗ F h 1 ∗ ⋯ ∗ F h m + 1, where F g is the Milnor fiber of g and F h iis the Milnor fiber of h i . This implies that F f has the homotopy type of a bouquet of spheres S n + m . So we can define the Milnor number μ ( f ) as the number of spheres in that bouquet, as in the complex setting.