Relations between the leading terms of a polynomial automorphism

Let I be the ideal of relations between the leading terms of the polynomials defining an automorphism of Kn. In this paper, we prove the existence of a locally nilpotent derivation which preserves I. Moreover, if I is principal, i.e. I=(R), we compute an upper bound for deg2(R) for some degree function deg2 defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of K3 and deduce two elementary proofs of the Jung–van der Kulk Theorem about the tameness of automorphisms of K2.