On polynomial automorphisms of affine spaces

In the first part of this paper we prove some general results on the linearizability of algebraic group actions on $\mathbb A^n$. As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on $\mathbb A^n$ with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group $G$ on the $n$-dimensional affine space $\mathbb A^n$ over an algebraically closed field $k$ of characteristic zero is linearizable in each of the following cases: 1) $n=3$; 2) $n=4$ and $G$ is not a one- or two-dimensional torus. In particular, this means that $\operatorname{GL}_3(k)$ is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over $k$.