对合与雅可比猜想
Involutions and the Jacobian conjecture
摘要：
著名的雅可比猜想询问具有非零标量雅可比的$K[x,y]$（$K$是特征零域）的自同态$f$是否可逆。令$α$为$K[x,y]$上的交换对合：$α(x)= y$和$α(y)= x$。$K[x,y]$的$α$ -自同态$f$是$K[x,y]$的保持对合$α$：$f α= αf$的自同态.证明了如果$f$是$K[x,y]$的具有非零标量雅可比矩阵的$α$ -自同态，则$f$是可逆的.在此基础上，我们得到了更多的结果，这意味着一个给定的自同态$f$具有一个非零的标量雅可比矩阵和额外的条件，涉及对合，是可逆的。
英文摘要：
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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