Some Remarks on the Jacobian Conjecture and Dru{\.z}kowski mappings

TL;DR

This paper advances understanding of the Jacobian Conjecture by proving it for certain non-homogeneous mappings, establishing equivalent statements, and analyzing inverse degrees for specific homogeneous power linear Keller maps.

Contribution

It provides new conditions under which the Jacobian Conjecture holds and offers explicit inverse formulas and degree bounds for particular classes of polynomial maps.

Findings

Jacobian Conjecture holds for certain non-homogeneous power linear mappings.

Equivalent formulations of the Jacobian Conjecture in various dimensions are established.

Degree bounds for inverses of specific homogeneous power linear Keller maps are derived.

Abstract

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension $r\geq 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$, we give the inverse polynomial map under the condition that $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d^3$.