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

Abstract: We compute by hand all quadratic homogeneous polynomial maps $H$ and all Keller maps of the form $x + H$, for which ${\rm rk} J H = 3$, over a field of arbitrary characteristic. Furthermore, we use computer support to compute Keller maps of the form $x + H$ with ${\rm rk} J H = 4$, namely: $\bullet$ all such maps in dimension $5$ over fields with $\frac12$; $\bullet$ all such maps in dimension $6$ over fields without $\frac12$. We use these results to prove the following over fields of arbitrary characteristic: for Keller maps $x + H$ for which ${\rm rk} J H \le 4$, the rows of $J H$ are dependent over the base field.