Polynomial maps with nilpotent Jacobians in dimension three I

Abstract

In this paper, we first prove that $u,v,h$ are linearly dependent over $\mathbf{K}$ if $JH$ is nilpotent and $H$ has the form: $H=(u(x,y,z),v(u,h),h(x,y))$ with $H(0)=0$ or $H=(u(x,y),v(u,h),h(x,y,z))$ with $H(0)=0$. Then we classify polynomial maps of the form $H=(u(x,y),v(x,y,z),h(x,y))$ in the case that $JH$ is nilpotent and $({°}_{y}u,{°}_{y}h)\le 2$.

21 pages