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Vanishing cycles and monodromy of complex polynomials

In this paper we describe the trivial summand for monodromy around a fibre of a polynomial map C → C, generalising and clarifying work of Artal Bartolo, Cassou-Noguès and Dimca [2], who proved similar results under strong restrictions on the homology of the general fibre and singularities of the other fibres. They also showed a polynomial map f : C → C has trivial global monodromy if and only if it is “rational of simple type” in the terminology of Miyanishi and Sugie. We refine this result and correct the Miyanishi-Sugie classification of such polynomials, pointing out that there are also non-isotrivial examples.