Power linear Keller maps of dimension four
The author proves the following fact connected to the Jacobian conjecture and tame generators conjecture. Let \(K\) 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 \(F_i(X_1,\dots,X_n) =X_i+(A_i(X_1,\dots,X_n))^{d}\) where \(A_i\) are linear forms and \(d>1\). The main theorem is: If \(n=4\) and \(\text{Jac}F\in K^*\) then the above \(F\) is linearly triangularizable, i.e. there exists a linear invertible map \(\phi\) such that \(\phi\cdot F\cdot\phi^{-1}(X_1,\dots,X_n)=(X_1+p_1, \dots,X_n+p_n),\quad p_i\in K[X_{i+1},\dots,X_n].\) Hence \(F\) (as above) is tame and invertible. The author [\textit{C. C.-A. Cheng}, Proc. Am. Math. Soc. 129, No. 10, 2819-2822 (2001; Zbl 0991.14022)] proved the same result for \(n=3\) but under a slight general definition of power linear maps.
linearly triangularizable polynomial map
scientific article; zbMATH DE number 1758340