The real Jacobian conjecture for maps with one component having degree 6

TL;DR

This paper proves that polynomial maps in the plane with one component of degree 6 and a non-zero Jacobian determinant are injective, advancing the understanding of the real Jacobian conjecture.

Contribution

It establishes the injectivity of polynomial maps with a degree 6 component, confirming the real Jacobian conjecture for maps with one component degree less than 7.

Findings

Polynomial maps with a degree 6 component and non-zero Jacobian are injective.

Supports the real Jacobian conjecture for maps with one component degree under 7.

Extends previous results by confirming injectivity in this specific case.

Abstract

We show that if $F=(p,q):\mathbb R^2\to \mathbb R^2$ is a polynomial map such that the degree of $p$ is $6$ and whose Jacobian determinant is nowhere zero, then $F$ is injective. This together with previous works in the literature, guarantees the validity of the real Jacobian conjecture in the plane provided that one of the coordinate functions of the map has degree smaller than $7$.