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

TL;DR

This paper classifies quadratic homogeneous polynomial maps and Keller maps with Jacobian rank 3 or 4, using manual calculations and computer support, and proves a dependency property of the Jacobian rows over arbitrary fields.

Contribution

It provides a complete classification of quadratic homogeneous polynomial maps and Keller maps with Jacobian rank 3 or 4, including new computational results and a general dependency theorem.

Findings

Classified all quadratic homogeneous maps with Jacobian rank 3.

Computed Keller maps with Jacobian rank 4 in specific dimensions.

Proved rows of Jacobian are dependent over the base field for rank ≤ 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.