### abstract

- this paper the case n = 3 seems to be solved in the negative by Shestakov and Umirbaev. In their preprints (under review) [16] and [17] they give an algorithm for recognizing tame automorphisms in dimension three. As a consequence they obtain that the famous Nagata automorphism # of k[x, y, z] ([15]) given by #(x) = x #(y) = y + (xz + y )z #(z) = z is not tame. The first question which then comes to mind is: what could possibly be a better set of candidate generators for the automorphism group of k or more generally of ? The aim of this paper is to discuss and study such a set of automorphisms. To motivate the idea we look again at the Nagata automorphism #. The polynomial # := xz + y has the property that #(#) = # i.e. it is #-invariant. Therefore # is a C[#]-homomorphism of C[x, y, z], which viewed over C[#] is linear in x, y and z