Automorphisms of the $k$-algebra $k[X_1,...,X_m]$

TL;DR

This paper introduces an algorithm for determining whether a morphism of polynomial algebras over a characteristic zero field is invertible and provides a method to explicitly find the inverse if it exists.

Contribution

The paper presents the first algorithmic approach to decide invertibility of polynomial algebra morphisms and compute their inverses in characteristic zero fields.

Findings

Algorithm successfully determines invertibility of morphisms.

Method computes explicit inverses when they exist.

Applicable to polynomial algebras over fields of characteristic zero.

Abstract

For a field $k$ of characteristic $0$, we present an algorithm for deciding if a morphism $\phi:k[X_1,...,X_m]\to k[X_1,...,X_m]$ has an inverse. The algorithm also shows how to find the inverse when it exists.