Polynomial maps with nilpotent Jacobians in dimension three II

TL;DR

This paper classifies specific polynomial maps with nilpotent Jacobians in three dimensions, focusing on particular forms and degree constraints, advancing understanding of their structure and properties.

Contribution

It provides a comprehensive classification of polynomial maps with nilpotent Jacobians under various form and degree conditions in three dimensions.

Findings

Classified polynomial maps of form H=(u,v,h) with nilpotent Jacobian for degree constraints

Extended classification to maps of form H=(u,v,h) with different variable dependencies

Identified structural properties of polynomial maps under nilpotency conditions

Abstract

In the paper, we first classify all polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(\deg_yu,\deg_yh)\leq 3$, $H(0)=0$. Then we classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,u), h(x,y))$ in the case that $JH$ is nilpotent and $(\deg v(x,y,0),\deg h)\leq 3$, $H(0)=0$. Finally, we classify polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in certain conditions.