The generalization of the Morse theory presented by Goresky and MacPherson is a landmark that divided completely the topological and geometrical study of singular spaces. Let { \(X_t\}_t\) be a suitable family of germs at 0 of complete intersection varieties in \({\mathbb {C}}^n\) and \(\{f_t\}_t, \{g_t\}_t\) families of non-constant polynomial functions on \(X_t\) . If the germs \(X_t\) , \(X_t \cap f_t^{-1}(0)\) and \(X_t\cap f_t^{-1}(0) \cap g_t^{-1}(0)\) are non-degenerate, locally tame, complete intersection varieties, for each t, we prove that the difference of the Brasselet numbers, \(\textrm{B}_{f_t,X_t}(0)\) and \(\textrm{B}_{f_t,X_t\cap g_t^{-1}(0)}(0)\) , is related with the number of Morse critical points on the regular part of the Milnor fiber of \(f_t\) appearing in a morsefication of \(g_t\) , even in the case where \(g_t\) has a critical locus with arbitrary dimension. This result connects topological and geometric properties and allows us to determine some interesting formulae, mainly in terms of the combinatorial information from Newton polyhedra.
