Two notes on the Jacobian conjecture (Q1082385)

Let K be a field of characteristic zero, let \(F=(F_ 1,...,F_ n)\) denote the endomorphism of \(K[X_ 1,...,X_ n]\) defined by \(X_ i\mapsto F_ i\), where \(F_ 1,...,F_ n\in K[X_ 1,...,X_ n]\), and let \(J(F)=(\partial F_ i/\partial X_ j)\) denote the Jacobian matrix of this endomorphism. The Jacobian conjecture is that F is an automorphism if J(F) is an invertible matrix. Two special cases of the conjecture are proved. Theorem 1. Suppose J(F) is invertible and there is a polynomial \(F_{n+1}\) such that \(F_ 1,...,F_{n+1}\) generate \(K[X_ 1,...,X_ n]\) as a K-algebra. Then F is an automorphism. Theorem 4. Suppose J(F) is invertible, and let \(J(F)=J(F)^ 0+J(F)^ 1\), where \(J(F)^ 0\in GL(n,K)\) and \(J(F)^ 1\) has no linear part. If \(J(F)^ 1\) has rank less than or equal to one, then F is an automorphism.