Computations of Keller maps over fields with $\tfrac16$

TL;DR

This paper classifies specific Keller maps with homogeneous components over fields with 1/6, providing new results on their structure and dependencies, especially for degrees 3 and 4 in low dimensions.

Contribution

It offers a comprehensive classification of Keller maps with homogeneous parts over fields with 1/6, including new general results on homogeneous polynomial maps.

Findings

Classification of Keller maps with deg H=3 and rk JH ≤ 2

Classification of Keller maps with deg H=3 in dimension ≤ 4

Classification of Keller maps with deg H=4 in dimension ≤ 3

Abstract

We classify Keller maps $x + H$ in dimension $n$ over fields with $\tfrac16$, for which $H$ is homogeneous, and (1) deg $H = 3$ and rk $JH \le 2$; (2) deg $H = 3$ and $n \le 4$; (3) deg $H = 4$ and $n \le 3$; (4) deg $H = 4 = n$ and $H_1, H_2, H_3, H_4$ are linearly dependent over $K$. In our proof of these classifications, we formulate (and prove) several results which are more general than needed for these classifications. One of these results is the classification of all homogeneous polynomial maps $H$ as in (1) over fields with $\tfrac16$.