Rational Polynomials of Simple Type

A polynomial map f : C2 → C is rational if its generic fibre, and hence every fibre, is of genus zero. It is of simple type if, when extended to a morphism f̃ : X → P1 of a compactification X of C2, the restriction of f̃ to each curve C of the compactification divisor D = X − C2 is either degree 0 or 1. The curves C on which f̃ is non-constant are called horizontal curves, so one says briefly “each horizontal curve is degree 1”. The classification of rational polynomials of simple type gained some new interest through the result of Cassou-Nogues, Artal-Bartolo, and Dimca [4] that they are precisely the polynomials whose homological monodromy is trivial (it suffices that the homological monodromy at infinity be trivial by an observation of Dimca). A classification appeared in [12], but it is incomplete. It implicitly assumes trivial geometric monodromy (on page 346, lines 10-11). Trivial geometric monodromy implies isotriviality (generic fibres pairwise isomorphic) and turns out to be equivalent to it for rational polynomials of simple type. The classification in the non-isotrivial case was announced in the final section of [17]. The main purpose of this paper is to prove it. But we recently discovered that there are also isotrivial rational polynomials that are not in [12], so we have added a classification for the isotrivial case using our methods. This case can also be derived from Kaliman’s classification [9] of all isotrivial polynomials. The fact that his list includes rational polynomials of simple type that are not in [12] appears not to have been noticed before (it also includes rational polynomials not of simple type). In general, the classification of polynomial maps f : C2 → C is an open problem with extremely rich structure. One notable result is the theorem of Abhyankar-Moh and Suzuki [1, 23] which classifies all polynomials with one fibre isomorphic to C. The analogous result for the next simplest case, where one fibre is isomorphic to C∗, is open except in special cases where the