Some Properties of and Open Problems on Hessian Nilpotent Polynomials

TL;DR

This paper investigates properties of Hessian nilpotent polynomials, their inversion pairs, and related maps, contributing to the understanding of the Jacobian conjecture by exploring algebraic structures and proposing open problems.

Contribution

It presents new results on Hessian nilpotent polynomials and their inversion pairs, advancing the algebraic approach to the Jacobian conjecture and suggesting directions for future research.

Findings

Proved several properties of HN polynomials and their inversion pairs

Analyzed symmetric polynomial and formal maps related to HN polynomials

Proposed open problems for further exploration of Hessian nilpotent polynomials

Abstract

In the recent progress [BE1], [M], [Z1] and [Z2], the well-known Jacobian conjecture ([BCW], [E]) has been reduced to a problem on HN (Hessian nilpotent) polynomials (the polynomials whose Hessian matrix are nilpotent) and their (deformed) inversion pairs. In this paper, we prove several results on HN polynomials, their (deformed) inversion pairs as well as the associated symmetric polynomial or formal maps. We also propose some open problems for further study of these objects.