The homogeneous reduction of the Jacobian Conjecture

Let $n$ be a positive integer, and let $f:{\mathbb{C}}^n\rightarrow {\mathbb{C}}^n$ be a polynomial map with Jacobian $J(f)=1$. Suppose that, for each $i$, $f_i=x_i-h_i$, where $h_i:{\mathbb{C}}^n\rightarrow {\mathbb{C}}$ is homogeneous of degree $d$ for some $d\in {\mathbb{N}}$. The homogeneous reduction of the Jacobian Conjecture. Then $f$ is invertible with polynomial inverse. The source attributes this reduction to the cited overview article and later notes that the case $d=3$ implies the full Jacobian Conjecture.