A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)∗3, i.e. F(X)=(x1+(a11x1+⋯+a1nxn)3,…,xn+(an1x1+⋯+annxn)3), satisfies TrJ((AX)∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ⩽9 in the case .