Polynomial maps with nilpotent Jacobians in dimension three (Q891548)

In the Jacobian Conjecture, an important role is played by polynomial mappings \(H=(H_{1},\ldots ,H_{n})\), \(H_{i}\in \boldsymbol{K}[x_{1},\ldots ,x_{n}]\), \(\boldsymbol{K}\) -- a field of characteristic zero, such that the Jacobian matrix \(JH\) of \(H\) is nilpotent i.e. \((JH)^{n}=0\) for some \(n\in \mathbb{N}\). The authors classify very special classes of such \ maps \(H\) in the case \(n=3\). The first class consists of maps of the form \[ H=(u(x,y),v(x,y,z),h(x,y,v(x,y,z))) \] with restrictions: \(H(0)=0\), \(H_{1},H_{2},H_{3}\) are linearly independent over \(\boldsymbol{K}\), \(JH\) is nilpotent and \(\gcd (\deg _{y}u,\deg _{y}h)=1. \) The second class consists of maps of the form \[ H=(u(x,y),v(x,y,z),h(x,y,z)) \] with restrictions: \(H(0)=0\), \(H_{1},H_{2},H_{3}\) are linearly independent over \(\boldsymbol{K}\), \(JH\) is nilpotent, \(\deg _{z}h\leq 2\), \(\deg _{z}h<\deg _{z}v\) and \(\gcd (\deg _{y}u,\deg _{y}h)=1\).