  Given a polynomial endomorphism F of the n-dimensional affine space over a
field K, we define a sequence of polynomial endomorphisms of the affine space
associated to F. We call F nice if there exists an integer m such that the m-th
term of the sequence vanishes. Then F is invertible and its inverse may be
computed in terms of the endomorphisms in the sequence. In this paper we study
the class of nice polynomial automorphisms and obtain that this class is large
and includes triangulable automorphisms and all linear cubic homogeneous
polynomial automorphisms of nilpotence index up to 3. We prove as well that the
nicety property is invariant under linear conjugation and determine that seven
of the eight forms in Hubbers' classification of cubic homogeneous
automorphisms in dimension 4 are nice and moreover the eighth one is a
composition of nice maps.
