A note on the Jacobian Conjecture

TL;DR

This paper investigates Druzkowski mappings related to the Jacobian Conjecture, establishing rank bounds under certain trace conditions and proving the conjecture for dimensions up to 9 when specific diagonal conditions are met.

Contribution

It provides new rank bounds for Druzkowski mappings under trace conditions and verifies the Jacobian Conjecture in dimensions up to 9 for a class of these mappings.

Findings

Rank of A is bounded by half the sum of dimension and zero-diagonal count when trace condition holds.

Jacobian Conjecture verified for Druzkowski mappings in dimension ≤ 9 with non-zero diagonal entries.

Trace condition implies structural constraints on Druzkowski mappings.

Abstract

In this note, we show that, if the Druzkowski mappings $F(X)=X+(AX)^{*3}$, i.e. $F(X)=(x_1+(a_{11}x_1+...+a_{1n}x_n)^3,...,x_n+(a_{n1}x_1+...+a_{nn}x_n)^3)$, satisfies $TrJ((AX)^{*3})=0$, then $rank(A)\leq 1/2(n+\delta)$ where $\delta$ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension $\leq 9$ in the case $\prod_{i=1}^{n}a_{ii}\neq0$.