Involutions and the Jacobian conjecture

TL;DR

This paper investigates the Jacobian conjecture in two variables by studying endomorphisms that preserve an involution, showing that such involution-preserving endomorphisms with non-zero Jacobian are invertible.

Contribution

It extends previous results by providing new conditions involving involutions that ensure invertibility of endomorphisms with non-zero Jacobian.

Findings

Involutions help characterize invertible endomorphisms.

Preservation of involution combined with non-zero Jacobian implies invertibility.

Additional conditions involving involutions lead to new invertibility results.

Abstract

The famous Jacobian conjecture asks if an endomorphism $f$ of $K[x,y]$ ($K$ is a characteristic zero field) having a non-zero scalar Jacobian is invertible. Let $\alpha$ be the exchange involution on $K[x,y]$: $\alpha(x)= y$ and $\alpha(y)= x$. An $\alpha$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $\alpha$: $f \alpha= \alpha f$. It was shown that if $f$ is an $\alpha$-endomorphism of $K[x,y]$ having a non-zero scalar Jacobian, then $f$ is invertible. Based on this, we bring more results that imply that a given endomorphism $f$ having a non-zero scalar Jacobian and additional conditions involving involutions, is invertible.