On proper polynomial maps of C

Two proper polynomial maps f1, f2 : C 2 −→ C are said to be equivalent if there exist Φ1, Φ2 ∈ Aut(C) such that f2 = Φ2 ◦ f1 ◦ Φ1. We investigate proper polynomial maps of topological degree d ≥ 2 up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case d = 2. 0 Introduction Let f : C −→ C be a dominant polynomial map. We say that f is proper if it is closed and for every point y ∈ C the set f(y) is compact. The topological degree d of f is defined as the number of preimages of a general point. The semi-group of proper polynomial maps from C to C is not completely understood yet. It is known that these maps cannot provide any counterexample to the Jacobian Conjecture, see [BCW82, Theorem 2.1]. Nevertheless, it is worthwhile to study them from other points of view, for instance analyzing their dynamical behaviour; this investigation was recently started in [DS03], [DS08], [FJ07a] and [FJ07b]. In the present paper we do not consider any dynamical question but we try to generalize to arbitrary d ≥ 3 the following theorem, proved in [Lam05]. Theorem 0.1 (Lamy). Let f : C −→ C be a proper polynomial map of topological degree 2. Then there exist Φ1, Φ2 ∈ Aut(C) such that f = Φ2 ◦ f̃ ◦ Φ1, where f̃(x, y) = (x, y). We say that two proper polynomial maps f1, f2 : C 2 −→ C are equivalent if there exist Φ1, Φ2 ∈ Aut(C) such that f2 = Φ2 ◦ f1 ◦ Φ1. One immediately check that equivalent maps have the same topological degree. Therefore Theorem 0.1 says that when d = 2 there is just one equivalence class, namely that of f̃ . The aim of our work is to answer some questions that naturally arise from Lamy’s result. The first one, already stated in [Lam05], is the following: Question 0.2. Is every proper polynomial map f : C −→ C equivalent to some map of type (x, y) −→ (x, P (y))? The answer is negative, and a counterexample is provided already in degree 3 by the proper map f : C −→ C given by f(x, y) = (x, y + xy). ∗Partially supported by Progetto MIUR di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse and by GNSAGA INDAM. AMS MSC: 14R10 (Primary), 14E05, 20H15 (Secondary)