Automorphisms of the affine 3-space of degree 3

Abstract

In this article we give two explicit families of automorphisms of degree $\le 3$ of the affine $3$-space ${\mathbb{A}}^3$ such that each automorphism of degree $\le 3$ of ${\mathbb{A}}^3$ is a member of one of these families up to composition of affine automorphisms at the source and target; this shows in particular that all of them are tame. As an application, we give the list of all dynamical degrees of automorphisms of degree $\le 3$ of ${\mathbb{A}}^3$; this is a set of $3$ integers and $9$ quadratic integers. Moreover, we also describe up to compositions with affine automorphisms for $n\ge 1$ all morphisms ${\mathbb{A}}^3\to {\mathbb{A}}^n$ of degree $\le 3$ with the property that the preimage of every affine hyperplane in ${\mathbb{A}}^n$ is isomorphic to ${\mathbb{A}}^2$.

Prepublished version, 45 pages, comments are welcome

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