specializations diagrams. One of the questions which may arise, placed in the global setting of polynomial functions, is the following: [*how does the topology of the polynomial change when deforming it?*]{}

Since Broughton’s fundamental paper [@Bro] there has been an enormous interest in studying the variation of topology of the fibers of a given polynomial, partly due to the relation to the Jacobian Conjecture. In our paper, we study a polynomial by embedding it into a family. This gives a new viewpoint and non-trivial results occur even in cases where the topology at infinity of the original polynomial is constant (see §\[s:arnold\]).

It is well-known that an isolated singular point of a holomorphic function can be deformed into finitely many singularities and that the sum of their Milnor numbers equals the original one. A generic deformation is a Morsification and it can be used to define vanishing cycles, see e.g. the Appendix of Brieskorn’s paper [@Br].

Unlike the case of germs, deformations of polynomials do not conserve neither the total Milnor number, nor the number of vanishing cycles. While focusing on the behaviour at infinity, there is a way to define generic deformations of $f_0$, within the class of degree 