Involutions and the Jacobian conjecture

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 $α$ be the exchange involution on $K[x,y]$: $α(x)= y$ and $α(y)= x$. An $α$-endomorphism $f$ of $K[x,y]$ is an endomorphism of $K[x,y]$ that preserves the involution $α$: $f α= αf$. It was shown that if $f$ is an $α$-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.

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