Exponential maps of a polynomial ring in two variables

Abstract

Let k be a field. We show that the ring of invariants of every exponential map on k[x, y] is generated by the image of x or y under a composition of triangular automorphisms. From this we obtain two well-known theorems on k[x, y] with no restriction on the characteristic of k: the Rentschler–Miyanishi theorem and the Jung–van der Kulk theorem.