Quadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $rk JH=3$

TL;DR

This paper classifies quadratic homogeneous polynomial maps and Keller maps with Jacobian rank 3 over any field, showing they are tame automorphisms up to a square part in characteristic 2.

Contribution

It provides a complete classification of quadratic homogeneous maps with Jacobian rank 3 and proves their tameness, extending understanding of Keller maps in arbitrary characteristic.

Findings

All such maps are classified explicitly.

Keller maps with these properties are tame automorphisms.

Results hold over fields of any characteristic.

Abstract

We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.