Surjectivity of quotient maps for algebraic (C,+)-actions and polynomial maps with contractible fibres

In this paper, we establish two results concerning algebraic (C,+)-actions on C. First of all, let φ be an algebraic (C,+)-action on C3. By a result of Miyanishi, its ring of invariants is isomorphic to C[t1, t2]. If f1, f2 generate this ring, the quotient map of φ is the map F : C3 → C2, x 7→ (f1(x), f2(x)). By using some topological arguments, we prove that F is always surjective. Secondly we are interested in dominant polynomial maps F : C → Cn−1 whose connected components of their generic fibres are contractible. For such maps, we prove the existence of an algebraic (C,+)-action φ on C for which F is invariant. Moreover we give some conditions so that F (C[t1, .., tn−1]) is the ring of invariants of φ.