The degeneration of the boundary of the Milnor fibre to the link of complex and real non-isolated singularities

We study the boundary of the Milnor fibre of real analytic singularities $f: (\bR^m,0) \to (\bR^k,0)$,$m\geq k$, with an isolated critical value and the Thom $a_f$-property. We define the vanishing zone for $f$ and we give necessary and sufficient conditions for it to be a fibre bundle over the link of the singular set of $f^{-1}(0)$. In the case of singularities of the type $\fgbar: (\bC^n,0) \to (\bC,0)$withanisoaltedcriticalvalue,$f, g$ holomorphic, we further describe the degeneration of the boundary of the Milnor fibre to the link of $\text{fgbar}$. As a milestone, we also construct a L^e’s polyhedron for real analytic singularities of the type $\text{fgbar}:({\text{bC}}^2,0)\to (\text{bC},0)$ such that either $f$ or $g$ depends only on one variable.