On some nice polynomial automorphisms

TL;DR

This paper introduces the concept of 'nice' polynomial automorphisms, a class characterized by a terminating sequence of associated endomorphisms, and explores their properties, including invariance under conjugation and inclusion of various automorphism types.

Contribution

The paper defines the class of nice polynomial automorphisms, proves its largeness, and shows it includes triangulable and certain cubic homogeneous automorphisms, expanding understanding of polynomial automorphism structure.

Findings

Nice automorphisms form a large class including triangulable automorphisms.

The nicety property is invariant under linear conjugation.

Most cubic homogeneous automorphisms in dimension 4 are nice, with one being a composition of nice maps.

Abstract

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.