On the Monodromy of Complex Polynomials

Consider a polynomial function f : C → C with generic fiber F . Let Bf be the bifurcation set of f; hence f induces a smooth locally trivial fibration over C \ Bf . Then, for any integer q ≥ 0 and any coefficient ring R, there is an associated monodromy representation ρ(f )q : π1 ( C \ Bf , pt ) −→ Aut H̃q(F,R) ) in (reduced) homology. Going around a circle in C large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M∞(f )q . We show that these monodromy operators at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation. The role played by this decomposition is crucial since there are examples of polynomials C2 → C having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan