Invariance of Milnor numbers and topology of complex polynomials

Abstract

We give a global version of Le-Ramanujam μ-constant theorem for polynomials. Let \((f_t)\), \(t\in[0,1]\), be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number μ(t), the Milnor number at infinity λ(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the \(f_t\) is a constant, then the polynomials \(f_0\) and \(f_1\) are topologically equivalent. For \(n>3\) we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the \(f_t\) are all equivalent.