Remarks on the Jacobian conjecture (Q1355502)

Let \(F=(F_1,\dots,F_n),\;F_i\in\mathbb C[X_1,\dots,X_n] \) and \(\text{Jac}(F)\in\mathbb C^*\). Let \(M_i=m_{i0}(Y)+m_{i1}(Y)T+ \dots +m_{id_i}(Y)T^{d_i}_i \in \mathbb C[Y_1,\dots,Y_n,T] \) be the minimal polynomial of the algebraic dependence of \(F_1,\dots,F_n,X_i\) over \(\mathbb C\), i.e. the non-zero minimal irreducible polynomial such that \(M_i(F_1,\dots,F_n,X_i)=0\). Then the polynomials \(m_{i0}(Y),\dots,m_{id_i}(Y)\) have no common zero in \(\mathbb C^n\). From this theorem the author obtains the flatness of \(\mathbb C[F,X_i]\) over \(\mathbb C[F]\) for \(i=1,\dots,n\), and two known theorems that polynomial birational mappings of \(\mathbb C^n\) with \(\text{Jac}(F)\in\mathbb C^*\) and injective polynomial mappings of \(\mathbb C^n\) into \(\mathbb C^n\) are automorphisms.