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

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 $r\ge 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d \ge 2$,wegivetheinversepolynomialmapundertheconditionthat$JH^3=0$. We shall show that $\mathrm{deg}(F^{-1})\le d^k$ if $k\le 2$ and $J{H}^{k+1}=0$, but also give an example with $d=2$ and $J{H}^4=0$ such that $\mathrm{deg}(F^{-1})>d^3$.