Polynomial maps with nilpotent Jacobians in dimension three I

TL;DR

This paper classifies certain polynomial maps with nilpotent Jacobians in three dimensions, establishing linear dependence conditions and providing a detailed classification for specific degree constraints.

Contribution

It proves linear dependence of components under nilpotency and classifies polynomial maps with degree restrictions, advancing understanding of polynomial maps with nilpotent Jacobians.

Findings

Proves linear dependence of components when Jacobian is nilpotent.

Classifies polynomial maps with specific degree constraints.

Provides structural insights into polynomial maps with nilpotent Jacobians.

Abstract

In this paper, we first prove that $u,v,h$ are linearly dependent over ${\bf K}$ 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 $(\deg_yu,\deg_yh)\leq 2$.