Relations between the leading terms of a polynomial automorphism

TL;DR

This paper investigates the structure of relations between leading terms of polynomial automorphisms, establishing the existence of a preserving derivation, and providing bounds and classifications for these relations, with applications to automorphisms of low-dimensional affine spaces.

Contribution

It proves the existence of a locally nilpotent derivation preserving the ideal of relations and characterizes principal ideals, offering new insights into automorphisms of affine spaces.

Findings

Existence of a locally nilpotent derivation preserving the relation ideal.

Computed upper bounds for degrees of principal relation ideals.

Classified all principal relation ideals for automorphisms of $K^3$.

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 $K^{2}$.