Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two

TL;DR

This paper classifies cubic homogeneous polynomial maps with Jacobian matrices of rank at most two over fields with characteristic not 2 or 3, establishing invertibility and tameness conditions for associated Keller maps.

Contribution

It provides a complete classification of such polynomial maps and proves invertibility and tameness properties for Keller maps derived from them.

Findings

All cubic homogeneous polynomial maps with Jacobian rank ≤ 2 are classified.

Keller maps constructed from these are invertible.

Such Keller maps are tame if the dimension is not 4.

Abstract

Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map then $F$ is invertible, and furthermore $F$ is tame if the dimension $n\neq 4$.