Published online by Cambridge University Press: 30 May 2018
Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. 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 $n\neq 4$.
The first author has been supported by the Netherlands Organisation of Scientific Research (NWO). The second author has been partially supported by the NSF of China (grant nos. 11771176 and 11601146) and by the China Scholarship Council.