Invariance of the global monodromies in families of nondegenerate polynomials in two variables

We are interested in a global version of Lê-Ramanujam μ-constant theorem for polynomials. We consider an analytic family {fs}, s \in [0, 1], of complex polynomials in two variables, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber of fs is constant, then we show that the global monodromy fibrations of fs are all isomorphic, and that the degree of fs is constant (up to an algebraic automorphism of C2).