On cubic-linear polynomial mappings

TL;DR

This paper explores the relationship between cubic-homogeneous and cubic-linear polynomial mappings, establishing conditions for their invertibility and conjugability, and provides explicit formulas and examples to illustrate these connections.

Contribution

It clarifies the concept of pairing between cubic-homogeneous and cubic-linear maps and derives formulas linking their inverses and conjugations, advancing understanding in the Jacobian conjecture context.

Findings

Invertibility of one map implies invertibility of the paired map

Explicit formulas relate inverses and conjugations of paired maps

Provides examples of conjugable cubic-linear mappings

Abstract

In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of "pairing" and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontrivial examples of conjugable cubic-linear mappings are provided as an application.