A note on the real Jacobian conjecture in degree 7

Abstract

Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7+\beta x^{6}y$ for ${\alpha }^2+{\beta }^2\ne 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg}p=7$ and $\mathrm{deg}q$ is even and coprime with 7.

9 pages