Some Results On The Jacobian Conjecture And Polynomial Automorphisms

TL;DR

This paper investigates properties of polynomial maps related to the Jacobian Conjecture, demonstrating injectivity of certain homogeneous polynomials, pairing of specific polynomial maps, and establishing bounds on the inverse map degree.

Contribution

It provides new insights into the structure of polynomial automorphisms, especially Druzkowski maps, and refines bounds on the degree of their inverses.

Findings

Homogeneous polynomials satisfying the Jacobian condition are injective on lines through the origin.

F and G' are paired, with F being a Druzkowski map and G' a related cubic homogeneous polynomial.

A more precise bound for the degree of the inverse map F^{-1} was established.

Abstract

In this paper, we will first show that, the homogeneous polynomials which satisfy the Jacobian condition are injective on the lines that pass through the origin. Secondly, we will show that $F$ and $G'$ are paired, where $F$ is a Druzkowski map and $G'$ is a cubic homogeneous polynomial which related to $F$. Finally, we will find a more exactly bound for the degree of $F^{-1}$, where $F$ is a invertible map.