Quadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $3\le \mathrm{rk}JH\le 4$

Abstract

We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x+H$, for which $\mathrm{rk}JH=3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x+H$ with $\mathrm{rk}JH=4$, namely: $\bullet$ all such maps in dimension $5$ over fields with $\frac{1}{2}$; $\bullet$ all such maps in dimension $6$ over fields without $\frac{1}{2}$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x+H$ for which $\mathrm{rk}JH\le 4$, the rows of $JH$ are dependent over the base field.

31 pages