Power linear Keller maps of dimension three (Q2723475)

The author proves the following fact connected to the Jacobian conjecture and tame generators conjecture. NEWLINENEWLINENEWLINELet \(K\) be a field of characteristic zero. A polynomial map \(F=(F_1, \dots ,F_n): K^n\to K^n\) is called power linear iff \(F_i(X_1,\dots ,X_n) =X_i+(A_i(X_1,\dots ,X_n))^{d_i}\) where \(A_i\) are linear forms and \(d_i>1\). NEWLINENEWLINENEWLINEThe main theorem is: If \(n=3\) and \(\text{Jac}F\in K^*\) then \(F\) is linearly triangularizable, i.e. there exists a linear invertible map \(\varphi\) such that \(\varphi\circ F\circ\varphi^{-1}(X_1,\dots ,X_n)=(X_1+p_1, \dots ,X_n+p_n),\quad p_i\in K[X_{i+1},\dots ,X_n].\) NEWLINENEWLINENEWLINEAs an easy corollary the author obtains that \(F\) (as above) is tame and invertible.