Some polynomial maps with Jacobian rank two or three

TL;DR

This paper classifies specific polynomial maps with nilpotent Jacobian matrices of rank two or three, providing a detailed structure and generalization of such maps in three and higher dimensions.

Contribution

It offers a complete classification of polynomial maps with nilpotent Jacobian matrices of rank two or three, extending the structure to higher dimensions.

Findings

Classified polynomial maps of a specific form with nilpotent Jacobian.

Generalized the structure of these maps to higher dimensions.

Provided explicit forms and conditions for the maps.

Abstract

In the paper, we first classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $\deg_zv\leq 1$. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),\allowbreak b_3x_3+\cdots+b_nx_n+H_2^{(0)}(x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$.