On the Jacobian conjecture: Reduction of coefficients (Q1346098)

The author proves the following two particular cases of the jacobian conjecture: I. Let \(F : \mathbb{R}^ n \to \mathbb{R}^ n\) be a real polynomial map of the form \(F(X) = X - N(X)\), \(X = (X_ 1, \dots, X_ n)\), \(N\) contains no linear and constant terms and coefficients of \(N\) are nonnegative. If the jacobian of \(F\) is a nonzero constant, then \(F\) is an automorphism (even more, \(F\) is a stable tame automorphism). II. If the above theorem holds for every \(n\) and every \(F\) (of the above form) with non-positive coefficients, then the jacobian conjecture is true for all \(n\) and all fields of characteristic zero.