Relations between the leading terms of a polynomial automorphism

Abstract

Let I be the ideal of relations between the leading terms of the polynomials defining an automorphism of K n . 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 deg 2 ( R ) for some degree function deg 2 defined by the automorphism. As applications, we determine all the principal ideals of relations for automorphisms of K 3 and deduce two elementary proofs of the Jung–van der Kulk Theorem about the tameness of automorphisms of K 2 .