A new class of Nilpotent Jacobians in any dimension

TL;DR

This paper classifies a new class of nilpotent Jacobian polynomial maps in any dimension and proves their invertibility, providing evidence supporting the Jacobian Conjecture for these maps.

Contribution

It introduces a novel class of polynomial maps with nilpotent Jacobians in arbitrary dimensions and confirms their invertibility, advancing understanding of the Jacobian Conjecture.

Findings

Classified a new class of nilpotent Jacobian polynomial maps in any dimension.

Proved the invertibility of maps of the form X + H with nilpotent Jacobian.

Confirmed the Jacobian Conjecture for this specific class of maps.

Abstract

The classification of the nilpotent Jacobians with some structure has been an object of study because of its relationship with the Jacobian Conjecture. In this paper we classify the polynomial maps in dimension $n$ of the form $H = (u(x,y), u_2(x,y,x_3), \ldots, u_{n-1}(x,y,x_n), h(x,y))$ with $JH$ nilpotent. In addition we prove that the maps $X + H$ are invertible, which shows that for this kind of maps the Jacobian Conjecture is verified.