Triangularization properties of power linear maps and the Structural Conjecture

TL;DR

This paper investigates properties of power linear Keller maps, demonstrating that certain properties are not equivalent and exploring their hierarchical relationships, with positive results in low dimensions and ranks.

Contribution

It shows that two properties in the Structural Conjecture are not equivalent and clarifies the position of linear triangularizability among them.

Findings

One property is strictly stronger than the other.

Linear triangularizability is intermediate between the two properties.

Positive results obtained for small dimensions and Jacobian ranks.

Abstract

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Druzkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small dimensions and small Jacobian ranks.