Invariance of Milnor numbers and topology of complex polynomials

2002/01/15 by Arnaud Bodin
Mathematics · #math.AG #math.GT

paper · pdf

published as Commentarii Mathematici Helvetici (2003, vol. 78, 134-152) · 16 pages

arxiv created 2002/01/15 · arxiv updated 2009/11/30

Abstract

We give a global version of Le-Ramanujam mu-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, the Milnor number at infinity, 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 f₀ and f₁ 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.

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